Answer:
[tex]m=6[/tex]
Step-by-step explanation:
Exponent properties:
We can use exponent property [tex]a^{b^c}=a^{(b\cdot c)}[/tex] to solve this problem.
Rewrite [tex]8[/tex] as [tex]2^3[/tex], then apply exponent property [tex]a^{b^c}=a^{(b\cdot c)}[/tex] to simplify:
[tex]2^{3^5}=2^{2m+3},\\2^{15}=2^{2m+3}[/tex]
If [tex]a^b=a^c[/tex], then [tex]b=c[/tex], because of log property [tex]\log a^b=b\log a[/tex]. Using this log property, you can take the log of both sides and divide by [tex]\log a[/tex] to get [tex]b=c[/tex]
Therefore, we have:
[tex]15=2m+3[/tex]
Subtract 3 from both sides:
[tex]12=2m[/tex]
Divide both sides by 6:
[tex]m=\frac{12}{2}=\boxed{6}[/tex]
Alternative:
Given [tex]8^5=2^{2m+3}[/tex], to move the exponent down, we'll use log properties.
Start by simplifying:
[tex]\log 32,768=2^{2m+3}[/tex]
Take the log of both sides, then use log property [tex]\log a^b=b\log a[/tex] to move the exponent down:
[tex]\log(32,768)=\log 2^{2m+3},\\\log (32,768)=(2m+3)\log 2[/tex]
Divide both sides by [tex]\log2[/tex]:
[tex]2m+3=\frac{\log (32,768)}{\log(2)}[/tex]
Subtract 3 from both sides:
[tex]2m=\frac{\log (32,768)}{\log(2)}-3[/tex]
Divide both sides by 2:
[tex]m=\frac{\log (32,768)}{2\log(2)}-\frac{3}{2}=\boxed{6}[/tex]
