Trigonometry Examples

$8^{2 x} + 8^{x} - 42 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $8^{2 x}$ as ${(8^{x})}^{2}$ .

${(8^{x})}^{2} + 8^{x} - 42 = 0$

Step 1.2

Let $u = 8^{x}$ . Substitute $u$ for all occurrences of $8^{x}$ .

$u^{2} + u - 42 = 0$

Step 1.3

Factor $u^{2} + u - 42$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 1.3.2

Write the factored form using these integers.

$(u - 6) (u + 7) = 0$

Step 1.4

Replace all occurrences of $u$ with $8^{x}$ .

$(8^{x} - 6) (8^{x} + 7) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$8^{x} - 6 = 0$

$8^{x} + 7 = 0$

Step 3

Set $8^{x} - 6$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $8^{x} - 6$ equal to $0$ .

$8^{x} - 6 = 0$

Step 3.2

Solve $8^{x} - 6 = 0$ for $x$ .

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Step 3.2.1

Add $6$ to both sides of the equation.

$8^{x} = 6$

Step 3.2.2

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

$\ln (8^{x}) = \ln (6)$

Step 3.2.3

Expand $\ln (8^{x})$ by moving $x$ outside the logarithm.

$x \ln (8) = \ln (6)$

Step 3.2.4

Divide each term in $x \ln (8) = \ln (6)$ by $\ln (8)$ and simplify.

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Step 3.2.4.1

Divide each term in $x \ln (8) = \ln (6)$ by $\ln (8)$ .

$\frac{x \ln (8)}{\ln (8)} = \frac{\ln (6)}{\ln (8)}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $\ln (8)$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{x \ln (8)}{\ln (8)} = \frac{\ln (6)}{\ln (8)}$

Step 3.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (6)}{\ln (8)}$

Step 4

Set $8^{x} + 7$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $8^{x} + 7$ equal to $0$ .

$8^{x} + 7 = 0$

Step 4.2

Solve $8^{x} + 7 = 0$ for $x$ .

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Step 4.2.1

Subtract $7$ from both sides of the equation.

$8^{x} = - 7$

Step 4.2.2

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

$\ln (8^{x}) = \ln (- 7)$

Step 4.2.3

The equation cannot be solved because $\ln (- 7)$ is undefined.

Undefined

Step 4.2.4

There is no solution for $8^{x} = - 7$

No solution

Step 5

The final solution is all the values that make $(8^{x} - 6) (8^{x} + 7) = 0$ true.

$x = \frac{\ln (6)}{\ln (8)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (6)}{\ln (8)}$

Decimal Form:

$x = 0.86165416 \dots$