Logarithms

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Quick recall

• A logarithm asks: to what power must the base be raised?
• For real logarithms, bases are positive and not 1; arguments are positive.

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Formula Cards

1. Meaning of a logarithm

$$\log_a x=y\iff a^y=x$$

For real logs: a > 0, a ≠ 1, and x > 0.

2. Product rule

$$\log_a(xy)=\log_a x+\log_a y$$

3. Quotient rule

$$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$$

4. Power rule

$$\log_a(x^k)=k\log_a x$$

5. Change of base

$$\log_a x=\frac{\log_b x}{\log_b a}$$

The new base b must also be positive and not 1.

6. Special values

$$\log_a a=1,\qquad\log_a1=0$$

Worked Examples

Example 1

Question: Evaluate log₁₀(1000).

Step 1: Write 1000 as a power of 10.

$$1000=10^3$$

Step 2: Read the exponent.

$$\log_{10}1000=3$$

Answer: 3.

Example 2

Question: Evaluate log₂(32).

Step 1: Write 32 as a power of 2.

$$32=2^5$$

Step 2: Read the exponent.

$$\log_2 32=5$$

Answer: 5.

Example 3

Question: Simplify log₁₀(20)+log₁₀(5).

Step 1: Combine using the product rule.

$$\log_{10}20+\log_{10}5=\log_{10}(20\times5)$$

Step 2: Evaluate.

$$\log_{10}100=2$$

Answer: 2.

Example 4

Question: Evaluate log₃(81) using the power rule.

Step 1: Write 81 as a power of 3.

$$81=3^4$$

Step 2: Bring the exponent outside.

$$\log_3(3^4)=4\log_3 3=4$$

Answer: 4.

Drill

Try each question before opening its answer.

Question 1

Evaluate $\log_{10}10000$.

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  4

Question 2

Evaluate $\log_5 125$.

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  3

Question 3

Simplify $\log_{10}50+\log_{10}2$.

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  2

Question 4

Simplify $\log_{10}100-\log_{10}10$.

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  1

Question 5

Evaluate $\log_2 64$.

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