Trigonometry Examples

$4^{2 h - 1} = 16$

Step 1

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

$\ln (4^{2 h - 1}) = \ln (16)$

Step 2

Expand $\ln (4^{2 h - 1})$ by moving $2 h - 1$ outside the logarithm.

$(2 h - 1) \ln (4) = \ln (16)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(2 h - 1) \ln (4)$ .

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Step 3.1.1

Apply the distributive property.

$2 h \ln (4) - 1 \ln (4) = \ln (16)$

Step 3.1.2

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$2 h \ln (4) - \ln (4) = \ln (16)$

Step 4

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

$2 h \ln (4) - \ln (4) - \ln (16) = 0$

Step 5

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

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Step 5.1

Add $\ln (4)$ to both sides of the equation.

$2 h \ln (4) - \ln (16) = \ln (4)$

Step 5.2

Add $\ln (16)$ to both sides of the equation.

$2 h \ln (4) = \ln (4) + \ln (16)$

Step 6

Divide each term in $2 h \ln (4) = \ln (4) + \ln (16)$ by $2 \ln (4)$ and simplify.

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Step 6.1

Divide each term in $2 h \ln (4) = \ln (4) + \ln (16)$ by $2 \ln (4)$ .

$\frac{2 h \ln (4)}{2 \ln (4)} = \frac{\ln (4)}{2 \ln (4)} + \frac{\ln (16)}{2 \ln (4)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 h \ln (4)}{2 \ln (4)} = \frac{\ln (4)}{2 \ln (4)} + \frac{\ln (16)}{2 \ln (4)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{h \ln (4)}{\ln (4)} = \frac{\ln (4)}{2 \ln (4)} + \frac{\ln (16)}{2 \ln (4)}$

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

$\frac{h \ln (4)}{\ln (4)} = \frac{\ln (4)}{2 \ln (4)} + \frac{\ln (16)}{2 \ln (4)}$

Step 6.2.2.2

Divide $h$ by $1$ .

$h = \frac{\ln (4)}{2 \ln (4)} + \frac{\ln (16)}{2 \ln (4)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

Cancel the common factor.

$h = \frac{\ln (4)}{2 \ln (4)} + \frac{\ln (16)}{2 \ln (4)}$

Step 6.3.1.2

Rewrite the expression.

$h = \frac{1}{2} + \frac{\ln (16)}{2 \ln (4)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$h = \frac{1}{2} + \frac{\ln (16)}{2 \ln (4)}$

Decimal Form:

$h = 1.5$