Trigonometry Examples

$4^{2 x - 5} = 64^{3 x}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$4^{2 x - 5} = 4^{3 (3 x)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 x - 5 = 3 (3 x)$

Step 3

Solve for $x$ .

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Step 3.1

Multiply $3$ by $3$ .

$2 x - 5 = 9 x$

Step 3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.1

Subtract $9 x$ from both sides of the equation.

$2 x - 5 - 9 x = 0$

Step 3.2.2

Subtract $9 x$ from $2 x$ .

$- 7 x - 5 = 0$

Step 3.3

Add $5$ to both sides of the equation.

$- 7 x = 5$

Step 3.4

Divide each term in $- 7 x = 5$ by $- 7$ and simplify.

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Step 3.4.1

Divide each term in $- 7 x = 5$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{5}{- 7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{5}{- 7}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{- 7}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{7}$

Decimal Form:

$x = - 0.\overline{714285}$