Exponential and Logarithmic Functions

Log Properties

Apply the product rule log(xy) = log(x) + log(y) to compute log_2(4) + log_2(8) = log_2(32) = 5.

Example: log_2(4) + log_2(8) = log_2(4 · 8) = log_2(32). Since 2^2 = 4 and 2^3 = 8, the sum 2 + 3 = 5 = log_2(32).

Example

Use the product rule to combine logs and compute the value.

highlighted = computed this step

Step 1 — Product rule

Set up the expression.

log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN)=\log_b M+\log_b N

Step 2 — Evaluate logs

Evaluate the two logs: log base 2 of 4 is 2, and of 8 is 3.

log⁡2(4)=2log⁡2(8)=3\log_{ 2 }( 4 )= \hl{2} \quad \log_{ 2 }( 8 )= \hl{3}

Step 3 — Combine product

Use the product rule inside the log: 4 times 8 = 32.

4⋅8=324 \cdot 8 = \hl{32}

Step 4 — Add log values

Add the log values: 2 + 3 = 5.

2+3=52 + 3 = \hl{5}

Step 5 — Result

The combined log is log base 2 of 32, which equals 5.

log⁡2(32)=5\log_{ 2 }( 32 )= \hl{5}
log-properties The product rule states: log_b(xy) = log_b(x) + log_b(y). This converts multiplication inside the log to addition of logs, and is the basis for logarithmic simplification.