Logarithmic and Exponential Form

Simple overview on how to convert back and forth between exponential and logarithmic form.

Converting Exponentials to Logarithms

We can go back between exponential form and logarithmic form using the following

\[a^y=x \quad \Longleftrightarrow \quad \log_a(x)=y\]

The exponential form is on the left side of the conversion

\[a^y=x\]

Here, $a$ is the base, $y$ is the exponent, and $x$ is the result.

The logarithmic form is on the right side of the conversion

\[\log_a(x)=y\]

Here, $a$ is still the base, $x$ becomes the argument of the logarithm, and $y$ is the value of the logarithm.

Watch this section: goal and topic introduction at 0:00 and mapping overview at 0:12.

Exponential To Logarithmic Form ($\Longrightarrow$)

Consider the following exponential equation

\[3^4=81\]

Match this to the pattern in the conversion above

\[a^y=x\]

So the base is $a=3$, the exponent is $y=4$, and the result is $x=81$.

In logarithmic form, the base stays the base, the result becomes the argument, and the exponent becomes what the logarithm is set equal to

\[\log_3(81)=4\]

Watch this section: converting an exponential equation at 1:39.

Logarithmic To Exponential Form ($\Longleftarrow$)

Consider the following logarithmic equation

\[\log_5(125)=h\]

We can match it to the form that is on the right side of the conversion above

\[\log_a(x)=y\]

So the base is $a=5$, the argument is $x=125$, and the logarithm value is $y=h$.

Mapping these values back to the exponential form of the conversion we get

\[5^h=125\]

Watch this section: converting a logarithmic equation at 2:14.

Solving For The Unknown Exponent (Using Final Form of last example)

Once the logarithmic equation has been rewritten as

\[5^h=125\]

the question becomes a little clearer about what we are looking for: $5$ raised to what power equals $125$?

Since

\[5^3=125\]

we get

\[h=3\]

So converting back to the logarithmic equation we could replace the $h$ with $3$ and get

\[\log_5(125)=3\]

Watch this section: solving for the unknown at 2:50 and interpreting the logarithm at 3:39.

Timestamp Guide

Section	What is shown	Video
Goal	Introduce converting between logarithmic and exponential equations.	0:00
Exponential to logarithmic map	Show how $a^y=x$ becomes $\log_a(x)=y$.	0:12
Logarithmic to exponential map	Show how $\log_a(x)=y$ becomes $a^y=x$.	0:50
Exponential example	Convert $3^4=81$ into $\log_3(81)=4$.	1:39
Logarithmic example	Convert $\log_5(125)=h$ into $5^h=125$.	2:14
Unknown exponent	Use $5^3=125$ to find $h=3$.	2:50
Meaning of a logarithm	Read $\log_5(125)=h$ as an exponent question.	3:39
Basic identities	Connect $a^0=1$, $a^1=a$, and $a^k=a^k$ to log identities.	3:58

Related Logarithm Work

Use the logarithms topic page for more logarithm notes. After this conversion idea feels comfortable, the logarithmic power rule archive, logarithmic product rule archive, and logarithmic quotient rule archive are natural next practice pages.

You can also watch the JoeCMath logarithms playlist: open the playlist on YouTube.