Algebra Examples

$y = 4^{2 x} - 3$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 4^{2 x} - 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4^{2 x} - 3 = 0$ .

$4^{2 x} - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$4^{2 x} = 3$

Step 1.2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4^{2 x}) = \ln (3)$

Step 1.2.4

Expand $\ln (4^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (4) = \ln (3)$

Step 1.2.5

Divide each term in $2 x \ln (4) = \ln (3)$ by $2 \ln (4)$ and simplify.

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Step 1.2.5.1

Divide each term in $2 x \ln (4) = \ln (3)$ by $2 \ln (4)$ .

$\frac{2 x \ln (4)}{2 \ln (4)} = \frac{\ln (3)}{2 \ln (4)}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{2 x \ln (4)}{2 \ln (4)} = \frac{\ln (3)}{2 \ln (4)}$

Step 1.2.5.2.1.2

Rewrite the expression.

$\frac{x \ln (4)}{\ln (4)} = \frac{\ln (3)}{2 \ln (4)}$

Step 1.2.5.2.2

Cancel the common factor of $\ln (4)$ .

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Step 1.2.5.2.2.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{\ln (3)}{2 \ln (4)}$

Step 1.2.5.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (3)}{2 \ln (4)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (3)}{2 \ln (4)}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 4^{2 (0)} - 3$

Step 2.2

Simplify $4^{2 (0)} - 3$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $0$ .

$y = 4^{0} - 3$

Step 2.2.1.2

Anything raised to $0$ is $1$ .

$y = 1 - 3$

Step 2.2.2

Subtract $3$ from $1$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (3)}{2 \ln (4)}, 0)$

y-intercept(s): $(0, - 2)$

Step 4