CSCA MATHEMATICS · MODULE 02 · LESSON 02.6

Logarithmic Functions

Understand logarithms as inverse exponential relationships, learn the domain restrictions, and apply essential logarithmic identities and laws.

Lesson overview

Logarithms are an essential part of mathematics. They are closely related to exponential functions and provide a way to determine an unknown exponent.

For example, suppose we want to find the power to which we must raise

\[2\]

to obtain

\[8\]

We know that:

\[2^3=8\]

Therefore, the required exponent is

\[3\]

. Logarithms provide a systematic way to express and calculate this exponent.

Logarithms are used in mathematics, physics, engineering, computer science, and other fields. They help us solve exponential equations and describe quantities that grow or decrease exponentially.

Learning objectives

By the end of this lesson, students should be able to:

  • Define a logarithm and explain its relationship with exponents.

  • Convert between logarithmic and exponential forms.

  • Identify valid logarithmic bases and arguments.

  • Apply the product, quotient, and power rules.

  • Determine the domain, range, and monotonicity of logarithmic functions.

  • Interpret logarithmic graphs.

  • Solve elementary logarithmic equations.

  • Solve basic logarithmic inequalities while respecting base restrictions.

  • Explain the inverse relationship between logarithmic and exponential functions.

COURSE SECTION

Part 1 . Understanding logarithms

1.1 What is a logarithm?

A logarithm answers the question:

To what power must we raise a given base to obtain a particular number?

Consider:

\[2^3=8\]

The base is

\[2\]

the exponent is

\[3\]

and the result is

\[8\]

We can express the same relationship using a logarithm:

\[\log_2 8=3\]

Both expressions describe the same mathematical relationship.

In general:

\[\log_a x=y\]

means that:

\[a^y=x\]

Here:

  • \(a\) is the logarithm’s base.

  • \(x\) is the argument.

  • \(y\) is the logarithm’s value.

The logarithm tells us which exponent produces the argument when the base is raised to that exponent.

1.2 Converting logarithmic form to exponential form

The fundamental relationship is:

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

For real-valued logarithms, the conditions are:

\[a>0,\qquad a\neq1,\qquad x>0\]

Worked example

Convert the following logarithmic expression to exponential form:

\[\log_3 81=4\]

The base is

\[3\]

the exponent is

\[4\]

and the result is

\[81\]

Therefore:

\[3^4=81\]

Worked example

Convert the exponential equation to logarithmic form:

\[5^3=125\]

The base is

\[5\]

the result is

\[125\]

and the exponent is

\[3\]

Therefore:

\[\log_5 125=3\]

Practice table

Logarithmic form Exponential form Value
\(\log_2 8\) \(2^3=8\) \(3\)
\(\log_3 27\) \(3^3=27\) \(3\)
\(\log_5 25\) \(5^2=25\) \(2\)
\(\log_{10}1000\) \(10^3=1000\) \(3\)
\(\log_4 16\) \(4^2=16\) \(2\)

1.3 Basic logarithmic identities

Several logarithmic values can be calculated immediately using the definition.

Identity A: Logarithm of the base

Since:

\[a^1=a\]

we have:

\[\log_a a=1\]

For example:

\[\log_7 7=1\]

Identity B: Logarithm of one

For any valid base:

\[a^0=1\]

Therefore:

\[\log_a 1=0\]

For example:

\[\log_5 1=0\]

Identity C: Logarithm and exponent cancellation

For any valid base

\[a\]

and real exponent

\[x\]
\[\log_a(a^x)=x\]

For example:

\[\log_2(2^5)=5\]

Similarly:

\[a^{\log_a x}=x,\qquad x>0\]

For example:

\[3^{\log_3 10}=10\]

Identity summary

Identity Result Explanation
\(\log_a a\) \(1\) \(a^1=a\)
\(\log_a 1\) \(0\) \(a^0=1\)
\(\log_a(a^x)\) \(x\) Logarithm and exponent cancel.
\(a^{\log_a x}\) \(x\) Valid for \(x>0\).

COURSE SECTION

Part 2 . Conditions on logarithms

2.1 Restrictions on the base

For a logarithm with a real value, the base must satisfy:

\[a>0\]

and:

\[a\neq1\]

Thus, the complete base restriction is:

\[a>0,\qquad a\neq1\]

Why?

  • A positive base produces real exponential values for every real exponent.

  • A base equal to \(1\) cannot generate arbitrary positive numbers because \(1^x=1\) for every real \(x\).

For example,

\[\log_2 8\]

is valid, while

\[\log_1 8\]

is not a valid real logarithm.

2.2 Restrictions on the argument

The argument of a real logarithm must be strictly positive:

\[x>0\]

For example:

\[\log_2 8\]

is valid because the argument is positive.

However:

\[\log_2 0\]

is undefined, and:

\[\log_2(-4)\]

is also undefined in the real number system.

Worked example: Finding the domain

Find the domain of:

\[f(x)=\log_2(x-3)\]

The argument must be positive:

\[x-3>0\]

Add

\[3\]

to both sides:

\[x>3\]

Therefore, the domain is:

\[D=(3,\infty)\]

Notice that the endpoint is excluded because the argument cannot equal zero.

Worked example: A linear argument

Find the domain of:

\[f(x)=\log_5(2x+6)\]

The argument must be positive:

\[2x+6>0\]

Subtract

\[6\]
\[2x>-6\]

Divide by

\[2\] :
\[x>-3\]

Therefore:

\[D=(-3,\infty)\]

Domain restriction table

Function Required condition Domain
\(\log_2 x\) \(x>0\) \((0,\infty)\)
\(\log_3(x-4)\) \(x-4>0\) \((4,\infty)\)
\(\log_5(2x+6)\) \(2x+6>0\) \((-3,\infty)\)
\(\log_7(10-x)\) \(10-x>0\) \((-\infty,10)\)

COURSE SECTION

Part 3 . Logarithmic rules

Logarithmic rules allow us to simplify expressions without calculating every power directly.

These rules are especially useful when logarithms contain products, quotients, or powers.

3.1 Product rule

The logarithm of a product is the sum of the logarithms, provided the arguments are positive and the base is valid.

\[\log_a(MN)=\log_a M+\log_a N\]

where:

\[M>0,\qquad N>0,\qquad a>0,\qquad a\neq1\]

Worked example

Simplify:

\[\log_2 8+\log_2 4\]

Apply the product rule in reverse:

\[\log_2 8+\log_2 4=\log_2(8\cdot4)\]

Multiply:

\[\log_2 32\]

Since:

\[2^5=32\]

the result is:

\[\log_2 8+\log_2 4=5\]

Another example

Expand:

\[\log_3(9x)\]

Assume

\[x>0\]

Using the product rule:

\[\log_3(9x)=\log_3 9+\log_3 x\]

Since:

\[\log_3 9=2\]

we obtain:

\[\log_3(9x)=2+\log_3 x\]

3.2 Quotient rule

The logarithm of a quotient is the difference between the logarithms of its numerator and denominator.

\[\log_a\left(\frac{M}{N}\right)
=\log_a M-\log_a N\]

where

\[M>0\]
\[N>0\]
\[a>0\]

and

\[a\neq1\]

Worked example

Simplify:

\[\log_2 32-\log_2 4\]

Apply the quotient rule in reverse:

\[\log_2 32-\log_2 4
=\log_2\left(\frac{32}{4}\right)\]

Simplify the fraction:

\[\log_2 8\]

Since:

\[2^3=8\]

the answer is:

\[\log_2 32-\log_2 4=3\]

Another example

Expand:

\[\log_5\left(\frac{x}{25}\right)\]

Assume

\[x>0\]

Using the quotient rule:

\[\log_5\left(\frac{x}{25}\right)
=\log_5 x-\log_5 25\]

Since:

\[\log_5 25=2\]

we obtain:

\[\log_5\left(\frac{x}{25}\right)=\log_5 x-2\]

3.3 Power rule

The logarithm of a positive number raised to a power can be rewritten by moving the exponent in front of the logarithm.

\[\log_a(M^r)=r\log_a M\]

where

\[M>0\]
\[a>0\]

and

\[a\neq1\]

Worked example

Simplify:

\[\log_2(8^2)\]

Using the power rule:

\[\log_2(8^2)=2\log_2 8\]

Since:

\[\log_2 8=3\]

we obtain:

\[\log_2(8^2)=2(3)=6\]

Another example

Expand:

\[\log_3(x^4)\]

For

\[x>0\]

the power rule gives:

\[\log_3(x^4)=4\log_3 x\]

For a negative nonzero

\[x\]

the original argument

\[x^4\]

is still positive, so the more general real-valued identity is:

\[\log_3(x^4)=4\log_3|x|,\qquad x\neq0\]

This distinction matters when determining domains.

3.4 Combining logarithmic rules

The product, quotient, and power rules can be combined to simplify more complicated expressions.

Worked example

Expand:

\[\log_2\left(\frac{x^3y}{z^2}\right)\]

Assume

\[x>0\]
\[y>0\]

and

\[z>0\]

Apply the quotient rule:

\[\log_2\left(\frac{x^3y}{z^2}\right)
=\log_2(x^3y)-\log_2(z^2)\]

Apply the product rule:

\[=\log_2(x^3)+\log_2 y-\log_2(z^2)\]
Source boundary

The supplied lesson text ends during the combined logarithmic-rules example in Section 3.4. The remaining steps are provided below to complete that worked example only; no additional lesson sections have been invented.

Complete the final worked example

Apply the power rule, assuming x, y, z are positive:

\[\log_2(x^3)+\log_2 y-\log_2(z^2)\]
\[=3\log_2 x+\log_2 y-2\log_2 z\]
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