A Logarithm Is Just the Question an Exponent Answers
Read the logarithm as the index a base needs, convert freely between exponential and logarithmic form, evaluate logs straight from the definition, and solve for a missing base or argument.
What does a logarithm actually ask you to find?
You already know how to answer *what is *. The answer is .
A logarithm asks the same relationship backwards: *what power turns into *. The answer is , and the way you write that question is
Read it out loud as **the index that needs in order to reach . Nothing more is going on.
That is why the previous chapter had to come first. Every logarithm is an index wearing different clothes, so every statement about indices becomes a statement about logarithms and back again.
Three statements, one fact:**
-
-
- five is the power of two that gives thirty-two
The small letter is the base, and it stays the base in both forms. In the is the number doing the multiplying and the is the target — swapping them is the commonest error of the whole chapter, and reading the notation aloud prevents it.
This page covers the first part of the ICSE Class 9 Mathematics chapter on logarithms: the definition, converting between the two forms, evaluating a logarithm directly, and solving for whichever of the three quantities is missing.
A logarithm asks the same relationship backwards: *what power turns into *. The answer is , and the way you write that question is
Read it out loud as **the index that needs in order to reach . Nothing more is going on.
That is why the previous chapter had to come first. Every logarithm is an index wearing different clothes, so every statement about indices becomes a statement about logarithms and back again.
Three statements, one fact:**
-
-
- five is the power of two that gives thirty-two
The small letter is the base, and it stays the base in both forms. In the is the number doing the multiplying and the is the target — swapping them is the commonest error of the whole chapter, and reading the notation aloud prevents it.
This page covers the first part of the ICSE Class 9 Mathematics chapter on logarithms: the definition, converting between the two forms, evaluating a logarithm directly, and solving for whichever of the three quantities is missing.
Formula
How do you convert between exponential form and logarithmic form?
One definition carries the whole chapter.
The base sits under the log and at the bottom of the power. The index is what the log equals. The number is what you are taking the log of.
Worked conversions, both directions.
- becomes
- becomes
- becomes
- becomes
- becomes
- becomes
Check that last one, because it looks strange: , as required. **A base smaller than gives negative logarithms for numbers above — the fraction has to be inverted to grow.
Two values are free, whatever the base.** Putting and into the definition gives
So and need no working at all.
Now the three conditions, which are examined more often than students expect.
- ****: a power of a positive base is always positive, so no index can ever produce a negative answer. does not exist
- **: a negative base makes even and odd indices behave differently, and the function breaks
- **: since for every , the question *what index turns into has no answer, and what index turns into * has infinitely many
Those conditions are not decoration — they are the reason the definition has an arrow pointing both ways. Remove any one and the two forms stop being equivalent.
The base sits under the log and at the bottom of the power. The index is what the log equals. The number is what you are taking the log of.
Worked conversions, both directions.
- becomes
- becomes
- becomes
- becomes
- becomes
- becomes
Check that last one, because it looks strange: , as required. **A base smaller than gives negative logarithms for numbers above — the fraction has to be inverted to grow.
Two values are free, whatever the base.** Putting and into the definition gives
So and need no working at all.
Now the three conditions, which are examined more often than students expect.
- ****: a power of a positive base is always positive, so no index can ever produce a negative answer. does not exist
- **: a negative base makes even and odd indices behave differently, and the function breaks
- **: since for every , the question *what index turns into has no answer, and what index turns into * has infinitely many
Those conditions are not decoration — they are the reason the definition has an arrow pointing both ways. Remove any one and the two forms stop being equivalent.
How do you evaluate a logarithm straight from the definition?
**Set the logarithm equal to , convert to exponential form, write both sides to the same prime base, and equate the indices. Four steps, every time.
Worked example 1.** Evaluate .
Let , so , giving .
Worked example 2 — the answer need not be a whole number. Evaluate .
Let , so . Neither base is a power of the other, so send both to the prime base :
Check: , as required.
Worked example 3 — a surd base. Evaluate .
Let the value be , so , which gives and .
Check: , as required. A base below the target's own base makes the index bigger — it takes more steps to climb the same height.
Worked example 4 — a negative answer. Evaluate .
Check: , as required.
Worked example 5 — a decimal base. Evaluate .
Since and :
The pattern behind all five is that the log is positive when the base and the number sit on the same side of , and negative when they sit on opposite sides. was negative because is above while is below it. Predicting the sign before you calculate catches most slips instantly.
Worked example 1.** Evaluate .
Let , so , giving .
Worked example 2 — the answer need not be a whole number. Evaluate .
Let , so . Neither base is a power of the other, so send both to the prime base :
Check: , as required.
Worked example 3 — a surd base. Evaluate .
Let the value be , so , which gives and .
Check: , as required. A base below the target's own base makes the index bigger — it takes more steps to climb the same height.
Worked example 4 — a negative answer. Evaluate .
Check: , as required.
Worked example 5 — a decimal base. Evaluate .
Since and :
The pattern behind all five is that the log is positive when the base and the number sit on the same side of , and negative when they sit on opposite sides. was negative because is above while is below it. Predicting the sign before you calculate catches most slips instantly.
How do you find a missing base, number or value in a logarithm?
Convert to exponential form first. Whichever of the three letters is unknown, the equation that follows is one you can already solve.
The definition has three slots. A question can hide any one of them.
Case 1 — the value is unknown. Solve .
This is the easiest shape: .
Case 2 — a fractional index on the number. Solve .
Check: means , as required.
Case 3 — the base is unknown. Solve .
Converting gives , so . Note that this is a cube root, not an index — when the base is missing you end up taking a root rather than a power.
Case 4 — an unknown base with a fractional value. Solve .
Check: because , as required. Raising both sides to the reciprocal index is the move — the same trick you used in the indices chapter.
Case 5 — the unknown appears twice. Solve .
Now the check that this chapter demands and others do not. After solving, confirm that your answer keeps the number positive and the base legal. If gave , then and the answer stands. But an equation whose solution forced to be negative would have no solution at all, however clean the algebra looked. A logarithm question is not finished until the restrictions have been checked — that habit is examined directly in Class 11.
The definition has three slots. A question can hide any one of them.
Case 1 — the value is unknown. Solve .
This is the easiest shape: .
Case 2 — a fractional index on the number. Solve .
Check: means , as required.
Case 3 — the base is unknown. Solve .
Converting gives , so . Note that this is a cube root, not an index — when the base is missing you end up taking a root rather than a power.
Case 4 — an unknown base with a fractional value. Solve .
Check: because , as required. Raising both sides to the reciprocal index is the move — the same trick you used in the indices chapter.
Case 5 — the unknown appears twice. Solve .
Now the check that this chapter demands and others do not. After solving, confirm that your answer keeps the number positive and the base legal. If gave , then and the answer stands. But an equation whose solution forced to be negative would have no solution at all, however clean the algebra looked. A logarithm question is not finished until the restrictions have been checked — that habit is examined directly in Class 11.
Exam tip
What is the fastest way to keep logarithm notation straight in the exam?
Write the conversion line explicitly before every calculation. *Let , so .* That single line is worth a method mark and prevents the base-and-number swap.
- Say it aloud in words. is *the index that needs to reach *. The base is the small one, and the small one is the multiplier
- Reduce both sides to a prime base immediately — , , or . Almost every question in this chapter is built from , , , , , , or a power of ten
- Predict the sign first. Base and number on the same side of gives a positive answer; opposite sides gives a negative one. A wrong sign is then visible immediately
- **Write and as one-liners**, with no working. Spending three lines on suggests you have not seen the definition
- Check the restrictions at the end: the number must be positive and the base must be positive and not . Examiners ask for this in the state the conditions part of a question
- Keep the subscript low and clear. and are different things, and rushed handwriting makes them look the same
The misconception to name and avoid. is not . Test it: , while . There is no law for the logarithm of a sum — the laws you meet in Part 2 are all about products, quotients and powers.
- Say it aloud in words. is *the index that needs to reach *. The base is the small one, and the small one is the multiplier
- Reduce both sides to a prime base immediately — , , or . Almost every question in this chapter is built from , , , , , , or a power of ten
- Predict the sign first. Base and number on the same side of gives a positive answer; opposite sides gives a negative one. A wrong sign is then visible immediately
- **Write and as one-liners**, with no working. Spending three lines on suggests you have not seen the definition
- Check the restrictions at the end: the number must be positive and the base must be positive and not . Examiners ask for this in the state the conditions part of a question
- Keep the subscript low and clear. and are different things, and rushed handwriting makes them look the same
The misconception to name and avoid. is not . Test it: , while . There is no law for the logarithm of a sum — the laws you meet in Part 2 are all about products, quotients and powers.
Did you know
How can a logarithm tell you how many digits a huge number has?
Here is something you can work out with nothing but the definition.
A number between and has two digits, and its logarithm to base lies between and . A number between and has three digits, and its log lies between and . The whole-number part of a base-ten logarithm is one less than the digit count.
Now use it on a number nobody would write out. You know that , which is just above , so
Therefore , and has 31 digits. You have just counted the digits of a number with no calculator and no multiplication.
That is what logarithms were built to do: replace a hard multiplication with an easy addition. Before electronic calculators, every long multiplication in engineering and navigation was done by looking up two logarithms, adding them, and looking the answer back up — and the tables at the back of your textbook are the remains of that method.
The same compression is why logarithmic scales exist. Sound is measured in decibels, earthquake energy on a logarithmic magnitude scale, and acidity as pH — a base-ten logarithm of hydrogen-ion concentration. Each step of on such a scale is a multiplication of the underlying quantity by ten, which is why a pH of is ten times as acidic as a pH of and not slightly more so.
The useful habit here is remembering what the scale is doing: when a quantity varies over an enormous range, taking its logarithm turns that range into something you can plot on a page.
A number between and has two digits, and its logarithm to base lies between and . A number between and has three digits, and its log lies between and . The whole-number part of a base-ten logarithm is one less than the digit count.
Now use it on a number nobody would write out. You know that , which is just above , so
Therefore , and has 31 digits. You have just counted the digits of a number with no calculator and no multiplication.
That is what logarithms were built to do: replace a hard multiplication with an easy addition. Before electronic calculators, every long multiplication in engineering and navigation was done by looking up two logarithms, adding them, and looking the answer back up — and the tables at the back of your textbook are the remains of that method.
The same compression is why logarithmic scales exist. Sound is measured in decibels, earthquake energy on a logarithmic magnitude scale, and acidity as pH — a base-ten logarithm of hydrogen-ion concentration. Each step of on such a scale is a multiplication of the underlying quantity by ten, which is why a pH of is ten times as acidic as a pH of and not slightly more so.
The useful habit here is remembering what the scale is doing: when a quantity varies over an enormous range, taking its logarithm turns that range into something you can plot on a page.
Exam relevance
Why do JEE and NEET questions keep coming back to logarithms?
This chapter is foundation work whose payoff arrives in three different papers.
Where it leads in Mathematics. Logarithms feed directly into the Class 11 chapter of the same name and then into Limits, Derivatives and Integrals, where and are among the standard functions whose derivatives you must know. JEE Main sets logarithmic equations as short items, and they nearly always start with the step you are practising here — converting to exponential form, or rewriting to a common base.
Where it leads in Chemistry. pH, and the relationship between equilibrium constants and free energy are all base-ten or natural logarithms. Both NEET and JEE Main ask numerical questions where the only difficulty is the log manipulation; candidates who can move fluently between and finish those in seconds.
Where it leads in Physics. Logarithmic scales for intensity and the log form of exponential decay both appear in Class 11 and 12, and graph questions frequently ask you to read a slope off a log plot.
Question types to expect. Evaluate a logarithm with an awkward base; solve for an unknown base or argument; identify which of four given logarithmic statements is valid. That last type is pure condition-checking — is the argument positive, is the base legal — which is exactly the restriction habit built above.
The single trap that costs marks. Forgetting that the argument must be positive. A solved equation can produce a value that makes the argument zero or negative, and that value must be rejected, not reported. JEE Advanced sets questions where two roots emerge and only one survives the domain check, and the whole mark sits on rejecting the other.
Board versus competitive emphasis. The ICSE paper wants the conversion line, the working and the stated conditions. A competitive paper wants the surviving root. Practise the long form now — the domain discipline it builds is the thing you will actually be tested on later.
Where it leads in Mathematics. Logarithms feed directly into the Class 11 chapter of the same name and then into Limits, Derivatives and Integrals, where and are among the standard functions whose derivatives you must know. JEE Main sets logarithmic equations as short items, and they nearly always start with the step you are practising here — converting to exponential form, or rewriting to a common base.
Where it leads in Chemistry. pH, and the relationship between equilibrium constants and free energy are all base-ten or natural logarithms. Both NEET and JEE Main ask numerical questions where the only difficulty is the log manipulation; candidates who can move fluently between and finish those in seconds.
Where it leads in Physics. Logarithmic scales for intensity and the log form of exponential decay both appear in Class 11 and 12, and graph questions frequently ask you to read a slope off a log plot.
Question types to expect. Evaluate a logarithm with an awkward base; solve for an unknown base or argument; identify which of four given logarithmic statements is valid. That last type is pure condition-checking — is the argument positive, is the base legal — which is exactly the restriction habit built above.
The single trap that costs marks. Forgetting that the argument must be positive. A solved equation can produce a value that makes the argument zero or negative, and that value must be rejected, not reported. JEE Advanced sets questions where two roots emerge and only one survives the domain check, and the whole mark sits on rejecting the other.
Board versus competitive emphasis. The ICSE paper wants the conversion line, the working and the stated conditions. A competitive paper wants the surviving root. Practise the long form now — the domain discipline it builds is the thing you will actually be tested on later.
Key takeaways
What should you be able to do with logarithms before Part 2?
Part 1 is one definition, used in every direction.
- **, where the base is the small letter in both forms
- Read it aloud** as *the index that needs in order to reach *, and the base-number swap becomes impossible
- ** and for every legal base, with no working required
- To evaluate**, set it equal to , convert, reduce both sides to a prime base and equate indices — answers may be fractional or negative
- To find a missing base, you take a root rather than a power; to find a missing number, you take a power
- The restrictions matter: , and , and a solution that breaks them must be rejected
- Predict the sign from whether the base and the number lie on the same side of
- **There is no law for ** — the laws in Part 2 handle products, quotients and powers only
Every one of these is a re-reading of the indices chapter, which is the most useful thing to carry forward. Cover the worked examples above, evaluate , and from scratch, and see whether the conversion line now writes itself.
- **, where the base is the small letter in both forms
- Read it aloud** as *the index that needs in order to reach *, and the base-number swap becomes impossible
- ** and for every legal base, with no working required
- To evaluate**, set it equal to , convert, reduce both sides to a prime base and equate indices — answers may be fractional or negative
- To find a missing base, you take a root rather than a power; to find a missing number, you take a power
- The restrictions matter: , and , and a solution that breaks them must be rejected
- Predict the sign from whether the base and the number lie on the same side of
- **There is no law for ** — the laws in Part 2 handle products, quotients and powers only
Every one of these is a re-reading of the indices chapter, which is the most useful thing to carry forward. Cover the worked examples above, evaluate , and from scratch, and see whether the conversion line now writes itself.