Solve for x 4(1+10^(5x))=9

$4 (1 + 10^{5 x}) = 9$

Step 1

Divide each term in $4 (1 + 10^{5 x}) = 9$ by $4$ and simplify.

Divide each term in $4 (1 + 10^{5 x}) = 9$ by $4$ .

$\frac{4 (1 + 10^{5 x})}{4} = \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (1 + 10^{5 x})}{4} = \frac{9}{4}$

Divide $1 + 10^{5 x}$ by $1$ .

$1 + 10^{5 x} = \frac{9}{4}$

Step 2

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

Subtract $1$ from both sides of the equation.

$10^{5 x} = \frac{9}{4} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$10^{5 x} = \frac{9}{4} - 1 \cdot \frac{4}{4}$

Combine $- 1$ and $\frac{4}{4}$ .

$10^{5 x} = \frac{9}{4} + \frac{- 1 \cdot 4}{4}$

Combine the numerators over the common denominator.

$10^{5 x} = \frac{9 - 1 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 1$ by $4$ .

$10^{5 x} = \frac{9 - 4}{4}$

Subtract $4$ from $9$ .

$10^{5 x} = \frac{5}{4}$

Step 3

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{5 x}) = \log (\frac{5}{4})$

Step 4

Expand the left side.

Expand $\log (10^{5 x})$ by moving $5 x$ outside the logarithm.

$5 x \log (10) = \log (\frac{5}{4})$

Logarithm base $10$ of $10$ is $1$ .

$5 x \cdot 1 = \log (\frac{5}{4})$

Multiply $5$ by $1$ .

$5 x = \log (\frac{5}{4})$

Step 5

Divide each term in $5 x = \log (\frac{5}{4})$ by $5$ and simplify.

Divide each term in $5 x = \log (\frac{5}{4})$ by $5$ .

$\frac{5 x}{5} = \frac{\log (\frac{5}{4})}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{\log (\frac{5}{4})}{5}$

Divide $x$ by $1$ .

$x = \frac{\log (\frac{5}{4})}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\log (\frac{5}{4})}{5}$

Decimal Form:

$x = 0.01938200 \dots$

Solve for x 4(1+10^(5x))=9

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