Converting Between Exponential and Log Form

The key relationship

Exponential and logarithmic forms say the same thing in different ways:

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

Read it as: "$b$ raised to $y$ gives $x$" is the same as "the log base $b$ of $x$ is $y$."

Example 1: Convert $5^3 = 125$ to log form

Step 1 — Identify the parts: Base $= 5$, exponent $= 3$, result $= 125$.

Step 2 — Apply the pattern: The exponent becomes the answer to the log:

$\log_5 125 = 3$

Read as: "What power of 5 gives 125? Answer: 3."

Example 2: Convert $\log_4 64 = 3$ to exponential form

Step 1 — Identify the parts: Base $= 4$, log value $= 3$, argument $= 64$.

Step 2 — Apply the pattern: The log value becomes the exponent:

$4^3 = 64$

Read as: "4 raised to the 3rd power equals 64."

The memory trick

In $b^y = x$, the three parts are: base, exponent (y), result (x). In $\log_b x = y$: the base stays the base, the result goes inside the log, and the exponent becomes the answer.

More examples to practice

• $2^5 = 32 \iff \log_2 32 = 5$
• $10^{-2} = 0.01 \iff \log_{10} 0.01 = -2$
• $e^0 = 1 \iff \ln 1 = 0$

Try it in the visualization

Adjust the base and exponent. Both forms are shown side by side with arrows mapping each part. The color-coding shows which piece in the exponential form corresponds to which piece in the log form.