Trigonometry Examples

$10^{2 x} - 3 \cdot 10^{x} - 18 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $10^{2 x}$ as ${(10^{x})}^{2}$ .

${(10^{x})}^{2} - 3 \cdot 10^{x} - 18 = 0$

Step 1.2

Let $u = 10^{x}$ . Substitute $u$ for all occurrences of $10^{x}$ .

$u^{2} - 3 u - 18 = 0$

Step 1.3

Factor $u^{2} - 3 u - 18$ 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 1.3.2

Write the factored form using these integers.

$(u - 6) (u + 3) = 0$

Step 1.4

Replace all occurrences of $u$ with $10^{x}$ .

$(10^{x} - 6) (10^{x} + 3) = 0$

Step 2

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

$10^{x} - 6 = 0$

$10^{x} + 3 = 0$

Step 3

Set $10^{x} - 6$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $10^{x} - 6$ equal to $0$ .

$10^{x} - 6 = 0$

Step 3.2

Solve $10^{x} - 6 = 0$ for $x$ .

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Step 3.2.1

Add $6$ to both sides of the equation.

$10^{x} = 6$

Step 3.2.2

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

$\log (10^{x}) = \log (6)$

Step 3.2.3

Expand the left side.

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Step 3.2.3.1

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

$x \log (10) = \log (6)$

Step 3.2.3.2

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

$x \cdot 1 = \log (6)$

Step 3.2.3.3

Multiply $x$ by $1$ .

$x = \log (6)$

Step 4

Set $10^{x} + 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $10^{x} + 3$ equal to $0$ .

$10^{x} + 3 = 0$

Step 4.2

Solve $10^{x} + 3 = 0$ for $x$ .

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$10^{x} = - 3$

Step 4.2.2

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

$\log (10^{x}) = \log (- 3)$

Step 4.2.3

The equation cannot be solved because $\log (- 3)$ is undefined.

Undefined

Step 4.2.4

There is no solution for $10^{x} = - 3$

No solution

Step 5

The final solution is all the values that make $(10^{x} - 6) (10^{x} + 3) = 0$ true.

$x = \log (6)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \log (6)$

Decimal Form:

$x = 0.77815125 \dots$