Exponential and Logarithmic Functions

Log Definition

Understand logarithms via the definition: log_b(x) = y means b^y = x. Verify log_2(8) = 3 since 2^3 = 8.

Example: log_2(8) = 3 because 2^3 = 8. Verify: raise the base 2 to the power 3 to recover 8.

Example

Connect logarithmic form and exponential form using the same values.

highlighted = computed this step

Step 1 — Definition

Set up the expression.

log⁡b(x)=y  ⟺  by=x\log_b(x)=y\iff b^y=x

Step 2 — Log form

Read the log statement: log base 2 of 8 equals 3.

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

Step 3 — Exponential form

Rewrite it as exponential form: 2^3 = 8.

23=8\hl{2} ^{ \hl{3} }= \hl{8}

Step 4 — Verified

Both forms agree, so the value is 3.

log⁡2(8)=3\log_{ 2 }( 8 )= \hl{3}
log-definition The logarithm log_b(x) answers: "to what power must b be raised to get x?" Formally, log_b(x) = y if and only if b^y = x (b > 0, b ≠ 1, x > 0).