What You'll Learn in This Section
Welcome to Chapter 2! In this first topic, we will bridge the gap between exponential equations (powers) and logarithms. Logarithms allow us to solve for exponents, making them the direct inverse operation of exponentiation.
- Inverse of Exponents: Understanding $b^x = a \iff \log_b(a) = x$.
- John Napier & Henry Briggs History: How logs simplified scientific multiplication.
- Logarithms with Base 10: Working with powers of 10 and fractional decimals.
- Formal Definition: The constraints on base $b$ ($b > 0$, $b \neq 1$) and input $a$ ($a > 0$).
Start Learning
Study the concepts below, then work through the Exercise 2.1 Solutions, grab the Worksheets, review the Revision Notes, and finish with the Interactive Tests.
Introduction to Logarithms: Concepts
Origin of Logarithms
Imagine a world before computers and calculators, when scientists had to do astronomical calculations containing lengthy multiplication and division of large numbers. It took tremendous time and effort.
Then came John Napier (1550–1617), a Scottish mathematician, with a revolutionary idea: turning multiplication into addition, and division into subtraction. He introduced logarithms, a computational tool that simplified complex arithmetic.
Later, English mathematician Henry Briggs refined this idea, establishing common logarithms (base 10), which became standard in science and engineering tables.
In a music studio, a speaker produces a sound that is $1000$ times more intense than the softest sound a human ear can hear. Instead of writing "$1000$ times," scientists write:
Sound Level = $\log_{10}(1000) = 3$
Why $3$? Because $10^3 = 1000$.
The Inverse of Exponents
We already know about exponents. For example: $2^3 = 8$ Here, the Base is 2, the Exponent is 3, and the Resultant is 8.
Now, let's ask a different question: "2 raised to what power gives 8?"
That power is 3. We express this in logarithmic form as:
$\log_2(8) = 3$
This is read as: "logarithm of 8 to the base 2 is 3."
Logarithms through Powers of 10
Let us look at how positive and negative powers of 10 convert into logarithms:
Positive Powers
| Exponential | Logarithmic |
|---|---|
| $10^0 = 1$ | $\log_{10}(1) = 0$ |
| $10^1 = 10$ | $\log_{10}(10) = 1$ |
| $10^2 = 100$ | $\log_{10}(100) = 2$ |
| $10^3 = 1000$ | $\log_{10}(1000) = 3$ |
| $10^4 = 10000$ | $\log_{10}(10000) = 4$ |
Negative Powers
| Exponential | Logarithmic |
|---|---|
| $10^{-1} = 0.1$ | $\log_{10}(0.1) = -1$ |
| $10^{-2} = 0.01$ | $\log_{10}(0.01) = -2$ |
| $10^{-3} = 0.001$ | $\log_{10}(0.001) = -3$ |
| $10^{-4} = 0.0001$ | $\log_{10}(0.0001) = -4$ |
Formal Definition of Logarithm
For any positive base $b$ (where $b > 0$ and $b \neq 1$) and any positive number $a$, if: $b^x = a$ then: $\log_b(a) = x$
- If $b = 1$, then $1^x = a$. Since $1$ raised to any power is always 1, we couldn't evaluate things like $\log_1(5)$, which is mathematically undefined.
- If $b < 0$, raising negative bases to fractional powers (like $b^{1/2} = \sqrt{b}$) results in complex numbers, which are excluded from real algebra.
(a) Logarithmic form for $3^{-2} = \frac{1}{9}$
(b) Exponential form for $\log_5(625) = 4$
Show Solution
Since $b = 3$, $x = -2$, and $a = \frac{1}{9}$:
$\log_3\left(\frac{1}{9}\right) = -2$
(b) Exponential form:
Since base is $5$, exponent is $4$, and result is $625$:
$5^4 = 625$
Show Solution
$\log_2(1) = 0$
Exercise 2.1 — Ideal Textbook Solutions
Question 1
Write equivalent logarithmic statements
(a) $5^3 = 125$
(b) $2^5 = 32$
(c) $7^1 = 7$
(d) $3^{-1} = \frac{1}{3}$
View Complete Solution & Explanation
Recall the transformation: $b^x = a \iff \log_b(a) = x$.
(a) Solution:
Here, base $b = 5$, exponent $x = 3$, result $a = 125$.
$\log_5(125) = 3$
(b) Solution:
Here, base $b = 2$, exponent $x = 5$, result $a = 32$.
$\log_2(32) = 5$
(c) Solution:
Here, base $b = 7$, exponent $x = 1$, result $a = 7$.
$\log_7(7) = 1$
(d) Solution:
Here, base $b = 3$, exponent $x = -1$, result $a = \frac{1}{3}$.
$\log_3\left(\frac{1}{3}\right) = -1$
Question 2
Write equivalent exponential statements
(a) $\log_2(16) = 4$
(b) $\log_9(81) = 2$
(c) $\log_5\left(\frac{1}{5}\right) = -1$
(d) $\log_2(1) = 0$
View Complete Solution & Explanation
Recall the transformation: $\log_b(a) = x \iff b^x = a$.
(a) Solution:
Base $b = 2$, exponent $x = 4$, result $a = 16$.
$2^4 = 16$
(b) Solution:
Base $b = 9$, exponent $x = 2$, result $a = 81$.
$9^2 = 81$
(c) Solution:
Base $b = 5$, exponent $x = -1$, result $a = \frac{1}{5}$.
$5^{-1} = \frac{1}{5}$
(d) Solution:
Base $b = 2$, exponent $x = 0$, result $a = 1$.
$2^0 = 1$
Question 3
Evaluate Logarithmic Values
(a) $\log_{10}(1000)$
(b) $\log_6(36)$
(c) $\log_2(64)$
View Complete Solution & Explanation
To evaluate $\log_b(a) = x$, we solve the exponential equation $b^x = a$.
(a) Solution:
Let $\log_{10}(1000) = x$.
$\Rightarrow 10^x = 1000 = 10^3 \Rightarrow x = 3$.
$\log_{10}(1000) = 3$
(b) Solution:
Let $\log_6(36) = x$.
$\Rightarrow 6^x = 36 = 6^2 \Rightarrow x = 2$.
$\log_6(36) = 2$
(c) Solution:
Let $\log_2(64) = x$.
$\Rightarrow 2^x = 64 = 2^6 \Rightarrow x = 6$.
$\log_2(64) = 6$
Worksheets — Logarithms & Exponents
Exponential to Logarithmic Conversions
Focuses on roster mappings, simple base conversions, and integer evaluations.
-
1
Convert the following exponential statements into logarithmic form:
(a) $2^6 = 64$ (b) $10^{-3} = 0.001$ (c) $4^3 = 64$ (d) $7^0 = 1$ -
2
Convert the following logarithmic statements into exponential form:
(a) $\log_3(81) = 4$ (b) $\log_{10}(0.01) = -2$ (c) $\log_5(125) = 3$ -
3
Find the value of:
(a) $\log_2(32)$ (b) $\log_7(49)$ (c) $\log_3\left(\frac{1}{27}\right)$
Fractional Bases & Decimal Logs
Focuses on solving for variables and fractional roots.
-
4
Find the value of $x$ in each of the following equations:
(a) $\log_4(x) = 3$ (b) $\log_x(36) = 2$ (c) $\log_5(x) = -2$ -
5
Evaluate the following fractional base logarithms:
(a) $\log_{1/2}(8)$ (b) $\log_{1/3}\left(\frac{1}{9}\right)$ (c) $\log_{2/3}\left(\frac{8}{27}\right)$ -
6
State the exact constraints on base $b$ and value $a$ for the term $\log_b(a)$ to be mathematically defined. Why is $\log_1(5)$ undefined?
Nested Logs & Variable Bases
Solving complex systems of equations using basic definitions.
-
7
Evaluate the following nested logarithmic expressions:
(a) $\log_2(\log_3(81))$ (b) $\log_{10}(\log_2(1024))$ -
8
Solve for $x$ if:
(a) $\log_x(125) = 3$ (b) $\log_{\sqrt{2}}(16) = x$ (c) $\log_{x}(0.0001) = -4$ -
9
Let $y = \log_b(a)$. If $a$ is doubled while $b$ is kept constant, does $y$ double? Show with an example (e.g., $b=2$, $a=4$). Explain why.
Quick Revision Notes — Logarithms & Exponents
60-Second Summary
The core takeaways in under a minute
Logarithms represent the inverse of exponentiation. If you know a base raised to a power gives a result ($b^x = a$), the logarithm tells you that power ($\log_b(a) = x$). John Napier founded logarithms to make calculations easier, later refined by Henry Briggs into base-10 systems. The base $b$ must be positive ($b > 0$) and not equal to $1$ ($b \neq 1$), while the input number $a$ must also be strictly positive ($a > 0$). Common base-10 values include $\log_{10}(1000) = 3$ and $\log_{10}(0.01) = -2$.
Memory Mnemonics & Rules
Tricks to remember conversion layouts
The base of the exponent is the base of the logarithm. When converting, keep it small and low. $b^x = a \quad \longleftrightarrow \quad \log_{b}(a) = x$
Conversion Matrix Cheat-Sheet
Standard conversions to memorize
| Exponential Form | Logarithmic Form | Value type / Properties |
|---|---|---|
| $b^1 = b$ | $\log_b(b) = 1$ | Log of base itself is always 1 |
| $b^0 = 1$ | $\log_b(1) = 0$ | Log of 1 to any valid base is always 0 |
| $10^3 = 1000$ | $\log_{10}(1000) = 3$ | Positive integer power |
| $10^{-2} = 0.01$ | $\log_{10}(0.01) = -2$ | Negative decimal power |
| $2^5 = 32$ | $\log_2(32) = 5$ | Binary base evaluation |
Common Mistakes to Avoid
Watch out for these classic exam traps!
Converting $\log_2(8) = 3$ into $3^2 = 8$. Always remember that the base is 2, so it is $2^3 = 8$.
Writing $\log_b(-5) = x$. Since a positive base $b$ raised to any real power is always positive ($b^x > 0$), logarithms of negative numbers or zero are strictly undefined in real algebra.
Common logarithm uses base 10 ($\log_{10}$). Natural logarithm uses base $e \approx 2.718$ ($\log_e$ or $\ln$). Make sure you check which base is requested.
Logarithms & Exponential Form Chapter Tests
Take a Practice Test
Select your testing level to practice exponential to logarithmic form conversions, base definitions, and fractional power evaluations. Each test is out of 24 Marks and contains 10 structured questions. Compare your responses with marking keys to self-grade descriptive parts.
Basic Test
Direct exponential conversions, base-10 logs, and simple integer values.
Standard Test
Solving logarithmic variables, fractional base values, and mathematical base definitions.
Advanced Test
Double nested logs, proof transformations, quadratic logarithms, and John Napier history.