11th Grade Q1

Finding the zeros of an exponential function means finding the input values where the output is zero.

An exponential function has the form:

where:

• $a$ is a constant multiplier,
• $b$ is the base of the exponential,
• $x$ is the variable (exponent),
• $c$ is a constant shift.

The goal is to solve for $x$ when $f(x)=0$.

Method

1. Set the function equal to zero

To find the zeros, set the function equal to $0$:

2. Isolate the exponential term

Rearrange the equation to isolate the term involving $b^x$. Subtract $c$ from both sides:

Then, divide both sides by $a$:

3. Check for validity

For $b^x$ to be valid, $b^x$ must be positive, since exponential functions with a positive base never produce negative values. If $-\frac{c}{a}\le 0$, the equation has no zeros.

If $-\frac{c}{a}>0$, continue to the next step.

4. Solve for $x$

Take the logarithm of both sides to solve for $x$. You can use either natural logarithms ($\ln$) or base-$b$ logarithms. Using natural logarithms:

This gives the $x$-value of the zero, provided it exists.

Example

Let’s find the zeros of $f(x)=3\cdot 2^x-6$.

1. Set $f(x)=0$:
   

1. Isolate $2^x$:
   

1. Solve for $x$:
   

The zero is at $x=1$.

Key Points to Remember

• Exponential functions usually have no zeros if $-\frac{c}{a}\le 0$.
• When zeros exist, use logarithms to solve for $x$.