To solve the equation [tex]\( 5 \cdot 4^x = 320 \)[/tex] for [tex]\( x \)[/tex], we can follow these steps:
1. Isolate the exponential term:
Divide both sides of the equation by 5 to isolate [tex]\( 4^x \)[/tex]:
[tex]\[
\frac{5 \cdot 4^x}{5} = \frac{320}{5}
\][/tex]
Simplifying, we get:
[tex]\[
4^x = 64
\][/tex]
2. Express 64 as a power of 4:
Notice that 64 can be written as a power of 4. Indeed, [tex]\( 64 = 4^3 \)[/tex]. Thus, we re-write the equation:
[tex]\[
4^x = 4^3
\][/tex]
3. Use the property of exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
x = 3
\][/tex]
Therefore, the solution to the equation [tex]\( 5 \cdot 4^x = 320 \)[/tex] is
[tex]\[
x = 3
\][/tex]
