Solve for x 4*2^(5x)=21

$4 \cdot 2^{5 x} = 21$

Simplify.

Rewrite $4$ as $2^{2}$ .

$2^{2} \cdot 2^{5 x} = 21$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{2 + 5 x} = 21$

Divide each term in $2^{2 + 5 x} = 21$ by $4$ .

$\frac{2^{2 + 5 x}}{4} = \frac{21}{4}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2^{2 + 5 x} = 21$

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

$\ln (2^{2 + 5 x}) = \ln (21)$

Expand $\ln (2^{2 + 5 x})$ by moving $2 + 5 x$ outside the logarithm.

$(2 + 5 x) \ln (2) = \ln (21)$

Simplify the left side.

Apply the distributive property.

$2 \ln (2) + 5 x \ln (2) = \ln (21)$

Reorder $2 \ln (2)$ and $5 x \ln (2)$ .

$5 x \ln (2) + 2 \ln (2) = \ln (21)$

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

$5 x \ln (2) + 2 \ln (2) - \ln (21) = 0$

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

Subtract $2 \ln (2)$ from both sides of the equation.

$5 x \ln (2) - \ln (21) = - 2 \ln (2)$

Add $\ln (21)$ to both sides of the equation.

$5 x \ln (2) = - 2 \ln (2) + \ln (21)$

Divide each term in $5 x \ln (2) = - 2 \ln (2) + \ln (21)$ by $5 \ln (2)$ and simplify.

Divide each term in $5 x \ln (2) = - 2 \ln (2) + \ln (21)$ by $5 \ln (2)$ .

$\frac{5 x \ln (2)}{5 \ln (2)} = \frac{- 2 \ln (2)}{5 \ln (2)} + \frac{\ln (21)}{5 \ln (2)}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x \ln (2)}{5 \ln (2)} = \frac{- 2 \ln (2)}{5 \ln (2)} + \frac{\ln (21)}{5 \ln (2)}$

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 2 \ln (2)}{5 \ln (2)} + \frac{\ln (21)}{5 \ln (2)}$

Cancel the common factor of $\ln (2)$ .

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 2 \ln (2)}{5 \ln (2)} + \frac{\ln (21)}{5 \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (2)}{5 \ln (2)} + \frac{\ln (21)}{5 \ln (2)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $\ln (2)$ .

Cancel the common factor.

$x = \frac{- 2 \ln (2)}{5 \ln (2)} + \frac{\ln (21)}{5 \ln (2)}$

Rewrite the expression.

$x = \frac{- 2}{5} + \frac{\ln (21)}{5 \ln (2)}$

Move the negative in front of the fraction.

$x = - \frac{2}{5} + \frac{\ln (21)}{5 \ln (2)}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{5} + \frac{\ln (21)}{5 \ln (2)}$

Decimal Form:

$x = 0.47846348 \dots$

Solve for x 4*2^(5x)=21

Related Questions

Support LCMGCF.com by disabling your adblocker.