Question

The exponential equations 8^x = 480 does not have an integer solution. But it lies between two integer values. Fill in the blanks with two integer values. The number 8 must be raised to a power between the numbers blank and blank to equal 480

Answer 1

To find the two integer values between which the solution for the equation \( 8^x = 480 \) lies, we can first express \( 8 \) as a power of \( 2 \):

\[
8 = 2^3
\]

Thus, we can rewrite the equation as:

\[
(2^3)^x = 480
\]

This simplifies to:

\[
2^{3x} = 480
\]

To solve for \( x \), we can take the logarithm (base 2) of both sides:

\[
3x = \log_2(480)
\]

\[
x = \frac{\log_2(480)}{3}
\]

Now, we can find \( \log_2(480) \) by changing the base to either base 10 or using known values.

First, let's find \( \log_2(480) \). We know:

\[
480 = 2^5 \times 3 \times 5
\]

Calculating \( \log_2(480) \) gives:

\[
\log_2(480) = \log_2(2^5) + \log_2(3) + \log_2(5)
\]
\[
= 5 + \log_2(3) + \log_2(5)
\]

Using approximate values:
- \( \log_2(3) \approx 1.585 \)
- \( \log_2(5) \approx 2.3219 \)

So,

\[
\log_2(480) \approx 5 + 1.585 + 2.3219 \approx 8.9069
\]

Now, substituting back in for \( x \):

\[
x \approx \frac{8.9069}{3} \approx 2.96897
\]

This value indicates that \( x \) lies between \( 2 \) and \( 3 \). We can confirm this by evaluating \( 8^2 \) and \( 8^3 \):

\[
8^2 = 64
\]
\[
8^3 = 512
\]

Since \( 64 < 480 < 512 \), it confirms that \( x \) lies between \( 2 \) and \( 3 \).

Thus, the two integer values are:

**2 and 3.**