Precalculus Examples

$4 - 5 \ln (2 h - 1) = 7$

Step 1

Move all terms not containing $\ln (2 h - 1)$ to the right side of the equation.

Tap for more steps...

Step 1.1

Subtract $4$ from both sides of the equation.

$- 5 \ln (2 h - 1) = 7 - 4$

Step 1.2

Subtract $4$ from $7$ .

$- 5 \ln (2 h - 1) = 3$

Step 2

Divide each term in $- 5 \ln (2 h - 1) = 3$ by $- 5$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- 5 \ln (2 h - 1) = 3$ by $- 5$ .

$\frac{- 5 \ln (2 h - 1)}{- 5} = \frac{3}{- 5}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{- 5 \ln (2 h - 1)}{- 5} = \frac{3}{- 5}$

Step 2.2.1.2

Divide $\ln (2 h - 1)$ by $1$ .

$\ln (2 h - 1) = \frac{3}{- 5}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Move the negative in front of the fraction.

$\ln (2 h - 1) = - \frac{3}{5}$

Step 3

To solve for $h$ , rewrite the equation using properties of logarithms.

$e^{\ln (2 h - 1)} = e^{- \frac{3}{5}}$

Step 4

Rewrite $\ln (2 h - 1) = - \frac{3}{5}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{3}{5}} = 2 h - 1$

Step 5

Solve for $h$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $2 h - 1 = e^{- \frac{3}{5}}$ .

$2 h - 1 = e^{- \frac{3}{5}}$

Step 5.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 h - 1 = \frac{1}{e^{\frac{3}{5}}}$

Step 5.3

Add $1$ to both sides of the equation.

$2 h = \frac{1}{e^{\frac{3}{5}}} + 1$

Step 5.4

Divide each term in $2 h = \frac{1}{e^{\frac{3}{5}}} + 1$ by $2$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $2 h = \frac{1}{e^{\frac{3}{5}}} + 1$ by $2$ .

$\frac{2 h}{2} = \frac{\frac{1}{e^{\frac{3}{5}}}}{2} + \frac{1}{2}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 h}{2} = \frac{\frac{1}{e^{\frac{3}{5}}}}{2} + \frac{1}{2}$

Step 5.4.2.1.2

Divide $h$ by $1$ .

$h = \frac{\frac{1}{e^{\frac{3}{5}}}}{2} + \frac{1}{2}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Simplify each term.

Tap for more steps...

Step 5.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$h = \frac{1}{e^{\frac{3}{5}}} \cdot \frac{1}{2} + \frac{1}{2}$

Step 5.4.3.1.2

Combine.

$h = \frac{1 \cdot 1}{e^{\frac{3}{5}} \cdot 2} + \frac{1}{2}$

Step 5.4.3.1.3

Multiply $1$ by $1$ .

$h = \frac{1}{e^{\frac{3}{5}} \cdot 2} + \frac{1}{2}$

Step 5.4.3.1.4

Move $2$ to the left of $e^{\frac{3}{5}}$ .

$h = \frac{1}{2 e^{\frac{3}{5}}} + \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$h = \frac{1}{2 e^{\frac{3}{5}}} + \frac{1}{2}$

Decimal Form:

$h = 0.77440581 \dots$