04 Logarithmic Functions

Logarithms and inverse functions

A Logarithm answers the question: what exponent on a given base produces a specified positive number? Its definition is

with $b>0$, $b\ne1$, and $x>0$.

For example, $\log_2 8=3$ because $2^3=8$, $\log_{10}\left(\frac{1}{100}\right)=-2$ because $10^{-2}=\frac{1}{100}$, and $\log_5 1=0$ because $5^0=1$.

Inverse relationships

The Exponential function $f(x)=b^x$ and the logarithmic function $f^{-1}(x)=\log_b x$ are inverses. Therefore,

for every real number $x$, and

for $x>0$. Their graphs are reflections across the line $y=x$.

Domain and graph behavior

For $y=\log_b x$, the Domain is

and the range is

The graph has a vertical asymptote at $x=0$, crosses the $x$-axis at $(1,0)$, and has no $y$-intercept. If $b>1$, the function is increasing; if $0<b<1$, it is decreasing.

The Natural logarithm is the logarithm with base $e$:

where $e\approx2.71828$. Thus, $\ln(e^x)=x$ and $e^{\ln x}=x$ for $x>0$.

Properties and change of base

Logarithm rules are consequences of exponent rules and apply when the relevant arguments are positive.

Product, quotient, and power rules

The Product Property states

For example,

The quotient property is

and the power property is

For instance,

The rules can be used in reverse. If the arguments are positive, then

Nonproperties

Logarithms do not distribute over addition or subtraction. In general,

and

The product and quotient rules apply only to multiplication and division inside the logarithm.

Changing the base

The Change-of-Base Formula is

For example,

Takeaway: Products become sums, quotients become differences, and powers become coefficients; sums and differences inside logarithms cannot be split.

Solving logarithmic equations

Before solving, determine the Domain: every logarithm argument must be greater than zero. This condition prevents invalid values involving the logarithm of zero or a negative number.

A single logarithm

For

rewrite in exponential form:

For example,

becomes

so

The original argument is positive for this value, so it is valid.

Equal logarithms

For logarithms with the same base,

Thus,

gives

so $x=5$. Both arguments are positive at this value.

Combining logarithms

Consider

The domain requires $x>0$ and $x-3>0$, hence $x>3$. Combining the left side gives

so

Factoring gives

The candidates are $x=4$ and $x=-1$, but only $x=4$ satisfies $x>3$. Always check candidates in the original equation.

Solving exponential equations

Logarithms are especially useful when the unknown occurs in an exponent.

Same base

If both sides can be written with the same base, equate exponents. For example,

implies

so $x=3$.

Different bases

For

take natural logarithms:

Using the power property gives

and therefore

Shifted exponential equations

First isolate the exponential expression. To solve

add $5$ and divide by $3$:

Taking the natural logarithm gives

so

Takeaway: Isolate the exponential term before taking logarithms, then use the power property to bring the exponent down.

Logarithmic inequalities

A Logarithmic inequality is controlled by the base because the base determines whether the logarithmic function is increasing or decreasing.

Base greater than one

When $b>1$, the function is increasing and the inequality direction is preserved:

For

conversion to exponential form gives

The domain requires $x-1>0$, or $x>1$. Combining the conditions gives

Base between zero and one

When $0<b<1$, the function is decreasing and the inequality direction reverses. Consider

Conversion gives

The domain requires $x+2>0$, or $x>-2$, while the converted inequality requires $x<\frac{1}{9}-2=-\frac{17}{9}$. These conditions are incompatible because $-\frac{17}{9}<-2$, so there is no real solution.

Reliable procedure

1. State the positivity condition for every logarithm argument.

2. Determine whether the base is greater than $1$ or between $0$ and $1$.

3. Convert or compare expressions, reversing the inequality only when the base is between $0$ and $1$.

4. Intersect the result with the domain.

Takeaway: Check both monotonicity and domain before accepting a solution.

Applications to growth, decay, and scale

Logarithms allow exponential models to be solved for time and help represent quantities spanning very large or very small ranges.

Continuous growth and decay

A continuous model has the form

where $A_0$ is the initial amount, $k$ is the growth or decay constant, and $t$ is time. Solving for $t$ gives

and therefore

For decay written as $A(t)=A_0e^{-rt}$, where $r>0$,

Example: continuous population growth

Suppose a population starts at $500$ and grows continuously at $4\%$ per year. Its model is

To find when it reaches $800$, solve

After division and taking natural logarithms,

so

The population reaches $800$ after approximately $11.75$ years.

Half-life

For a decay model $A(t)=A_0e^{kt}$ with $k<0$, the Half-life $T$ satisfies

After dividing by $A_0$ and taking natural logarithms,

Since $\ln(1/2)=-\ln2$,

Logarithmic scales

A logarithmic scale represents multiplicative changes through additive distances. This makes logarithms useful for measurements such as sound intensity, acidity, and earthquake magnitude. The exact interpretation depends on the scale and its reference value.

Problem-solving checklist and summary

Use the following workflow to organize solutions:

1. Identify the domain. Require every logarithm argument to be positive.

2. Choose a useful form. Use the logarithm definition, exponential form, or logarithm properties as appropriate.

3. Apply properties carefully. Product, quotient, and power rules do not apply to sums or differences inside a logarithm.

4. For exponential equations, isolate the exponential expression before taking logarithms.

5. For logarithmic equations, convert a single logarithm to exponential form or combine logarithms before comparing arguments.

6. For inequalities, determine whether the base is greater than $1$ or between $0$ and $1$.

7. Check every candidate in the original equation or inequality.

8. Round only at the end when using a calculator.

The central ideas are:

• $\log_b x=y$ means $b^y=x$.

• The base satisfies $b>0$ and $b\ne1$.

• Every logarithm argument must be positive.

• Logarithms reverse exponential operations.

• Products become sums, quotients become differences, and powers become coefficients.

• The Natural logarithm is $\ln x=\log_e x$.

• The Change-of-Base Formula is $\log_b M=\frac{\ln M}{\ln b}$.

• Inequality direction is preserved for bases greater than $1$ and reversed for bases between $0$ and $1$.

• Logarithms help solve for unknown exponents and model growth, decay, half-life, and multiplicative scales.