Precalculus Examples

$3 (5^{x - 1}) = 21$

Step 1

Divide each term in $3 \cdot 5^{x - 1} = 21$ by $3$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $3 \cdot 5^{x - 1} = 21$ by $3$ .

$\frac{3 \cdot 5^{x - 1}}{3} = \frac{21}{3}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{3 \cdot 5^{x - 1}}{3} = \frac{21}{3}$

Step 1.2.1.2

Divide $5^{x - 1}$ by $1$ .

$5^{x - 1} = \frac{21}{3}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Divide $21$ by $3$ .

$5^{x - 1} = 7$

Step 2

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

$\ln (5^{x - 1}) = \ln (7)$

Step 3

Expand $\ln (5^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x - 1) \ln (5) = \ln (7)$

Step 4

Simplify the left side.

Tap for more steps...

Step 4.1

Simplify $(x - 1) \ln (5)$ .

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$x \ln (5) - 1 \ln (5) = \ln (7)$

Step 4.1.2

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$x \ln (5) - \ln (5) = \ln (7)$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (5) - \ln (5) - \ln (7) = 0$

Step 6

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 6.1

Add $\ln (5)$ to both sides of the equation.

$x \ln (5) - \ln (7) = \ln (5)$

Step 6.2

Add $\ln (7)$ to both sides of the equation.

$x \ln (5) = \ln (5) + \ln (7)$

Step 7

Divide each term in $x \ln (5) = \ln (5) + \ln (7)$ by $\ln (5)$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $x \ln (5) = \ln (5) + \ln (7)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (5)}{\ln (5)} + \frac{\ln (7)}{\ln (5)}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $\ln (5)$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (5)}{\ln (5)} + \frac{\ln (7)}{\ln (5)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (5)}{\ln (5)} + \frac{\ln (7)}{\ln (5)}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Cancel the common factor of $\ln (5)$ .

Tap for more steps...

Step 7.3.1.1

Cancel the common factor.

$x = \frac{\ln (5)}{\ln (5)} + \frac{\ln (7)}{\ln (5)}$

Step 7.3.1.2

Rewrite the expression.

$x = 1 + \frac{\ln (7)}{\ln (5)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \frac{\ln (7)}{\ln (5)}$

Decimal Form:

$x = 2.20906195 \dots$