11/14 11. Logarithmic Functions and Equations

Meaning and Evaluation

A logarithm answers the question: “To what exponent must the base be raised to produce the argument?” The fundamental equivalence is

Here, $b$ is the base, $y$ is the exponent, and $x$ is the argument. The base must satisfy $b>0$ and $b\neq 1$, while the argument must satisfy $x>0$.

For example,

because

To convert logarithmic form to exponential form, keep the base, move the logarithm's result to the exponent, and set it equal to the argument. Thus,

becomes

The inverse relationships are

and

when the expressions are defined.

Takeaway: A logarithm is an exponent, and converting between logarithmic and exponential forms is the central tool for evaluating and solving logarithmic expressions.

Special Logarithms and Change of Base

Two important special cases are the common logarithm and the natural logarithm.

• A common logarithm has base $10$: $\log(x)=\log_{10}(x)$.

• A natural logarithm has base $e$: $\ln(x)=\log_e(x)$, where $e\approx 2.71828$.

Examples include

because $10^3=1000$, and

because the natural logarithm and the exponential function with base $e$ are inverses.

When a calculator does not provide a key for a particular base, use the change-of-base formula:

For example,

Takeaway: Common and natural logarithms are standard bases, and the change-of-base formula makes any valid logarithm calculable.

Properties and Rewriting

For positive numbers $M$ and $N$, logarithms follow rules that translate multiplication, division, and powers into addition, subtraction, and multiplication.

• Product property:

  $$\log_b(MN)=\log_b(M)+\log_b(N).$$

• Quotient property:

  $$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N).$$

• Power property:

  $$\log_b(M^p)=p\log_b(M).$$

These properties support expansion and condensation. For example, expand

In the reverse direction,

Every argument must be positive. A logarithm of a sum cannot be separated using these rules:

Likewise,

Takeaway: Use logarithm properties only for products, quotients, and powers—not for sums or differences inside a single logarithm.

Graphs and Transformations

The parent logarithmic function is

where $b>0$ and $b\neq 1$. Its domain is $(0,\infty)$, its range is all real numbers, and its vertical asymptote is $x=0$. It crosses the $x$-axis at $(1,0)$, has no $y$-intercept, and includes the key points $(b,1)$ and $\left(\frac{1}{b},-1\right)$.

• If $b>1$, the graph is increasing.

• If $0<b<1$, the graph is decreasing.

• The graph is the reflection of $y=b^x$ across $y=x$, because logarithmic and exponential functions are inverses.

For $y=\log_2(x)$, convenient points come from writing $x=2^y$:

A transformed function has the form

The parameter $h$ shifts the graph horizontally and moves the vertical asymptote to $x=h$; $k$ shifts it vertically; $a$ stretches or compresses it vertically and reflects it across the $x$-axis when $a<0$. The domain comes from the argument condition:

For example, $f(x)=\log_3(x-2)+1$ has domain $(2,\infty)$, range all real numbers, vertical asymptote $x=2$, and an increasing graph.

Solving Logarithmic Equations

Choose a solution method based on the equation's structure.

One logarithm equals a constant

Convert directly to exponential form:

For

rewrite the equation as

Therefore,

The argument is $81$, which is positive, so the solution is valid.

Equal logarithms with the same base

Apply the one-to-one property:

For

set the arguments equal:

This gives $x=4$, and both original arguments equal $5$, so the result is valid.

Several logarithms with the same base

First state the domain, then combine the logarithms. Consider

The restrictions are $x>0$ and $x-2>0$, so $x>2$. Using the product property gives

Convert to exponential form:

Thus,

so the candidates are $x=4$ and $x=-2$. The domain condition rejects $x=-2$, leaving $x=4$.

The same process applies to natural logarithms. For example,

becomes

so $x=e^2+3$.

Domain Checks and Valid Solutions

The logarithm domain restriction is essential throughout every solution. Each logarithm argument in the original equation must be positive. A candidate that makes an argument equal to zero or less than zero is invalid, even if it solves an intermediate polynomial equation.

Use this checking procedure:

1. Write the positivity condition for every logarithm argument.

2. Solve the equation using conversion, logarithm properties, or the one-to-one property.

3. Test every candidate in the original equation.

4. Reject any extraneous solution that violates an original positivity condition.

For example, in an equation containing $\log_b(x-2)$, the restriction is

Therefore, any algebraic candidate with $x\leq 2$ must be discarded. This check is particularly important when combining logarithms produces a quadratic, because the quadratic may have roots outside the original logarithmic domain.

Final checklist:

• Is every base positive and different from $1$?

• Is every original logarithm argument positive?

• Were logarithm properties applied only to valid expressions?

• Were all candidates checked in the original equation?

Takeaway: Domain restrictions are not an afterthought; they determine which algebraic candidates are genuine solutions.