Solve for t 10^(2t)=9 ?

$10^{2 t} = 9$ ?

Step 1

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

$\log (10^{2 t}) = \log (9)$

Step 2

Expand the left side.

Expand $\log (10^{2 t})$ by moving $2 t$ outside the logarithm.

$2 t \log (10) = \log (9)$

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

$2 t \cdot 1 = \log (9)$

Multiply $2$ by $1$ .

$2 t = \log (9)$

Step 3

Divide each term in $2 t = \log (9)$ by $2$ and simplify.

Divide each term in $2 t = \log (9)$ by $2$ .

$\frac{2 t}{2} = \frac{\log (9)}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 t}{2} = \frac{\log (9)}{2}$

Divide $t$ by $1$ .

$t = \frac{\log (9)}{2}$

Simplify the right side.

Rewrite $\log (9)$ as $\log (3^{2})$ .

$t = \frac{\log (3^{2})}{2}$

Expand $\log (3^{2})$ by moving $2$ outside the logarithm.

$t = \frac{2 \log (3)}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$t = \frac{2 \log (3)}{2}$

Divide $\log (3)$ by $1$ .

$t = \log (3)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \log (3)$

Decimal Form:

$t = 0.47712125 \dots$