Training Exponentials & Logarithms Introduction to Logarithms
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Introduction to Logarithms

20 min Exponentials & Logarithms

Introduction to Logarithms

A logarithm answers the question: "to what power must the base be raised to produce this number?" The statement log base b of x equals y means the same thing as b to the y equals x. Logarithms are the inverse of exponentiation, just as subtraction is the inverse of addition.

The two most common bases are 10 (common logarithm, written log) and e ≈ 2.718 (natural logarithm, written ln). Understanding logarithms unlocks the ability to solve exponential equations — you simply "take the log" of both sides.

This lesson introduces logarithmic notation, evaluating simple logarithms, and the inverse relationship between logs and exponents.

Logarithms are the inverse of exponential functions. They answer the question: "What exponent do I need?"

Definition

$$\log_b(x) = y \quad \Longleftrightarrow \quad b^y = x$$

Read "$\log$ base $b$ of $x$ equals $y$." This means $b$ raised to the $y$ power gives $x$.

Common and Natural Logarithms
  • $\log(x) = \log_{10}(x)$ — the common logarithm (base 10)
  • $\ln(x) = \log_e(x)$ — the natural logarithm (base $e$)
Example 1

Convert $2^5 = 32$ to logarithmic form.

$$\log_2(32) = 5$$

Example 2

Evaluate $\log_3(81)$.

Ask: $3^? = 81$. Since $3^4 = 81$:

$$\log_3(81) = 4$$

Example 3

Evaluate $\log_5\left(\dfrac{1}{25}\right)$.

$5^? = \frac{1}{25} = 5^{-2}$. So $\log_5\left(\frac{1}{25}\right) = -2$.

Example 4

Evaluate $\log(1000)$.

$10^3 = 1000$, so $\log(1000) = 3$.

Example 5

Evaluate $\ln(e^5)$.

Since $\ln$ and $e^x$ are inverses: $\ln(e^5) = 5$.

Special Values

Key Facts
  • $\log_b(1) = 0$ for any valid base $b$  (since $b^0 = 1$)
  • $\log_b(b) = 1$  (since $b^1 = b$)
  • $\log_b(b^x) = x$  (logarithm undoes the exponential)
  • $b^{\log_b(x)} = x$  (exponential undoes the logarithm)
Example 6

Convert $\log_4(x) = 3$ to exponential form and solve.

$4^3 = x \implies x = 64$.

Practice Problems

1. Convert $3^4 = 81$ to logarithmic form.
2. Convert $\log_2(64) = 6$ to exponential form.
3. Evaluate $\log_4(16)$
4. Evaluate $\log(0.01)$
5. Evaluate $\ln(1)$
6. Evaluate $\log_2(128)$
7. Evaluate $\log_9(3)$
8. Evaluate $\log_5(625)$
9. Evaluate $\log_{1/2}(8)$
10. Solve $\log_3(x) = 5$
11. Solve $\log_x(49) = 2$
12. Evaluate $10^{\log(7)}$
13. Evaluate $\log_{25}(5)$
14. Evaluate $e^{\ln(4)}$
15. Evaluate $\log_2\left(\frac{1}{16}\right)$
Show Answer Key

1. $\log_3(81) = 4$

2. $2^6 = 64$

3. $2$

4. $-2$  ($10^{-2} = 0.01$)

5. $0$

6. $7$

7. $\frac{1}{2}$  ($9^{1/2} = 3$)

8. $4$

9. $-3$  ($(\frac{1}{2})^{-3} = 8$)

10. $x = 243$

11. $x = 7$

12. $7$

13. $\frac{1}{2}$

14. $4$

15. $-4$

Exponential Functions Properties of Logarithms