11th Grade Q1

To determine the zeros of a logarithmic function, you solve for the input value $x$ that makes the function equal to zero. The zero of a function is the point where the graph crosses the $x$-axis.

A logarithmic function typically has the form:

where:

• $a$ is a constant multiplier,
• $b$ is the base of the logarithm,
• $x+d$ is the argument of the logarithm,
• $c$ is a constant shift.

Method

1. Set $f(x)=0$

To find the zero, set the function equal to zero:

2. Isolate the logarithmic term

Rearrange the equation to isolate the logarithmic expression:

3. Rewrite in exponential form

Use the definition of a logarithm to rewrite the equation in exponential form:

4. Solve for $x$

Subtract $d$ from both sides to solve for $x$:

Example

Find the zeros of $f(x)=2\cdot {\log }_3(x+4)-6$.

1. Set $f(x)=0$

1. Isolate the logarithmic term

1. Rewrite in exponential form

1. Solve for $x$

Key Points to Remember

• The zero may be written as the point $(x,0)$, where $x$ is the solution to the equation $f(x)=0$.