Calculus Examples

$10^{x + 3} = 5 e^{7 - x}$

Step 1

Take the log of both sides of the equation.

$\ln (10^{x + 3}) = \ln (5 e^{7 - x})$

Step 2

Expand $\ln (10^{x + 3})$ by moving $x + 3$ outside the logarithm.

$(x + 3) \ln (10) = \ln (5 e^{7 - x})$

Step 3

Rewrite $\ln (5 e^{7 - x})$ as $\ln (5) + \ln (e^{7 - x})$ .

$(x + 3) \ln (10) = \ln (5) + \ln (e^{7 - x})$

Step 4

Expand $\ln (e^{7 - x})$ by moving $7 - x$ outside the logarithm.

$(x + 3) \ln (10) = \ln (5) + (7 - x) \ln (e)$

Step 5

The natural logarithm of $e$ is $1$ .

$(x + 3) \ln (10) = \ln (5) + (7 - x) \cdot 1$

Step 6

Multiply $7 - x$ by $1$ .

$(x + 3) \ln (10) = \ln (5) + 7 - x$

Step 7

Solve the equation for $x$ .

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Step 7.1

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

$(x + 3) \ln (10) - \ln (5) = 7 - x$

Step 7.2

Apply the distributive property.

$x \ln (10) + 3 \ln (10) - \ln (5) = 7 - x$

Step 7.3

Simplify the left side.

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Step 7.3.1

Simplify $x \ln (10) + 3 \ln (10) - \ln (5)$ .

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Step 7.3.1.1

Simplify each term.

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Step 7.3.1.1.1

Simplify $3 \ln (10)$ by moving $3$ inside the logarithm.

$x \ln (10) + \ln (10^{3}) - \ln (5) = 7 - x$

Step 7.3.1.1.2

Raise $10$ to the power of $3$ .

$x \ln (10) + \ln (1000) - \ln (5) = 7 - x$

Step 7.3.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (10) + \ln (\frac{1000}{5}) = 7 - x$

Step 7.3.1.3

Divide $1000$ by $5$ .

$x \ln (10) + \ln (200) = 7 - x$

Step 7.4

Add $x$ to both sides of the equation.

$x \ln (10) + \ln (200) + x = 7$

Step 7.5

Subtract $\ln (200)$ from both sides of the equation.

$x \ln (10) + x = 7 - \ln (200)$

Step 7.6

Factor $x$ out of $x \ln (10) + x$ .

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Step 7.6.1

Factor $x$ out of $x \ln (10)$ .

$x (\ln (10)) + x = 7 - \ln (200)$

Step 7.6.2

Raise $x$ to the power of $1$ .

$x (\ln (10)) + x = 7 - \ln (200)$

Step 7.6.3

Factor $x$ out of $x^{1}$ .

$x (\ln (10)) + x \cdot 1 = 7 - \ln (200)$

Step 7.6.4

Factor $x$ out of $x (\ln (10)) + x \cdot 1$ .

$x (\ln (10) + 1) = 7 - \ln (200)$

Step 7.7

Divide each term in $x (\ln (10) + 1) = 7 - \ln (200)$ by $\ln (10) + 1$ and simplify.

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Step 7.7.1

Divide each term in $x (\ln (10) + 1) = 7 - \ln (200)$ by $\ln (10) + 1$ .

$\frac{x (\ln (10) + 1)}{\ln (10) + 1} = \frac{7}{\ln (10) + 1} + \frac{- \ln (200)}{\ln (10) + 1}$

Step 7.7.2

Simplify the left side.

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Step 7.7.2.1

Cancel the common factor of $\ln (10) + 1$ .

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Step 7.7.2.1.1

Cancel the common factor.

$\frac{x (\ln (10) + 1)}{\ln (10) + 1} = \frac{7}{\ln (10) + 1} + \frac{- \ln (200)}{\ln (10) + 1}$

Step 7.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{\ln (10) + 1} + \frac{- \ln (200)}{\ln (10) + 1}$

Step 7.7.3

Simplify the right side.

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Step 7.7.3.1

Move the negative in front of the fraction.

$x = \frac{7}{\ln (10) + 1} - \frac{\ln (200)}{\ln (10) + 1}$

Step 7.7.3.2

Combine the numerators over the common denominator.

$x = \frac{7 - \ln (200)}{\ln (10) + 1}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 - \ln (200)}{\ln (10) + 1}$

Decimal Form:

$x = 0.51525777 \dots$