Question

8^5 = 2^2m+3 Solve m

Answer 1

Answer:

$m=6$

Step-by-step explanation:

Exponent properties:

We can use exponent property $a^{b^c}=a^{(b\cdot c)}$ to solve this problem.

Rewrite $8$ as $2^3$, then apply exponent property $a^{b^c}=a^{(b\cdot c)}$ to simplify:

$2^{3^5}=2^{2m+3},\\2^{15}=2^{2m+3}$

If $a^b=a^c$, then $b=c$, because of log property $\log a^b=b\log a$. Using this log property, you can take the log of both sides and divide by $\log a$ to get $b=c$

Therefore, we have:

$15=2m+3$

Subtract 3 from both sides:

$12=2m$

Divide both sides by 6:

$m=\frac{12}{2}=\boxed{6}$

Alternative:

Given $8^5=2^{2m+3}$, to move the exponent down, we'll use log properties.

Start by simplifying:

$\log 32,768=2^{2m+3}$

Take the log of both sides, then use log property $\log a^b=b\log a$ to move the exponent down:

$\log(32,768)=\log 2^{2m+3},\\\log (32,768)=(2m+3)\log 2$

Divide both sides by $\log2$:

$2m+3=\frac{\log (32,768)}{\log(2)}$

Subtract 3 from both sides:

$2m=\frac{\log (32,768)}{\log(2)}-3$

Divide both sides by 2:

$m=\frac{\log (32,768)}{2\log(2)}-\frac{3}{2}=\boxed{6}$