Logarithmic and Exponential Form

This page follows the JoeCMath video Master Converting Logarithms to Exponents (Both Directions). The video shows how the same relationship can be written in exponential form or logarithmic form, then uses that conversion to explain a few basic logarithm identities.

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Main Conversion

The video begins with the conversion pattern

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

The exponential form is

\[a^y=x.\]

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

The logarithmic form is

\[\log_a(x)=y.\]

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

The video treats the logarithmic form as another way to ask for the exponent.

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

Exponential To Logarithmic Form

The first worked conversion starts with

\[3^4=81.\]

Match this to the pattern

\[a^y=x.\]

So the base is $3$, the exponent is $4$, and the result is $81$.

In logarithmic form, the base stays the base, the result becomes the argument, and the exponent moves to the other side:

\[\log_3(81)=4.\]

The conversion is not a new rule to memorize; it is the same relationship written in the other direction.

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

Logarithmic To Exponential Form

The next worked conversion starts with

\[\log_5(125)=h.\]

This is in the form

\[\log_a(x)=y.\]

So the base is $5$, the argument is $125$, and the logarithm value is $h$.

Now convert back to exponential form:

\[5^h=125.\]

This is the same equation written so the unknown exponent is visible.

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

Solving For The Unknown Exponent

Once the logarithmic equation has been rewritten as

\[5^h=125,\]

the question becomes: $5$ raised to what power equals $125$?

Since

\[5^3=125,\]

we get

\[h=3.\]

So

\[\log_5(125)=3.\]

This is the main interpretation from the video: a logarithm gives the exponent needed to reach the argument from the base.

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

Basic Logarithm Identities

The final section uses the same conversion idea to explain three basic logarithm identities.

Because

\[a^0=1,\]

we have

\[\log_a(1)=0.\]

Because

\[a^1=a,\]

we have

\[\log_a(a)=1.\]

Because

\[a^k=a^k,\]

we have

\[\log_a(a^k)=k.\]

Each identity comes from converting between exponential form and logarithmic form. The base, result, and exponent are being renamed, not replaced with a separate trick.

Watch this section: basic logarithm rules and identities at 3:58.

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.