how to solve 5(2^(3x)) = 8

Steps too please

answer is ln 1.6 / 3 ln 2

solve 5(2^(3x)) = 8

5(2^(3x)) = 8
2^3x = 8/5
2^3x = 1.6
take logarithms of both sides
3x log 2 = log 1.6
Divide both sides by 3 log 2
x = log 1.6/(3 log 2)

thank you

You're welcome!

To solve the equation 5(2^(3x)) = 8, you can follow these steps:

Step 1: Divide both sides of the equation by 5 to isolate the exponential term:
(5(2^(3x)))/5 = 8/5
2^(3x) = 8/5

Step 2: Rewrite 8/5 as a power of 2:
2^(3x) = 2^(3 - 1)
2^(3x) = 2^2

Step 3: Since the bases (2) are the same on both sides, we can equate the exponents:
3x = 2

Step 4: Divide both sides of the equation by 3 to solve for x:
3x/3 = 2/3
x = 2/3

The answer is x = 2/3. However, this differs from the answer you provided (ln 1.6 / 3 ln 2). If you are looking for a solution involving natural logarithms, let's verify the solution:

Step 1: Rewrite the equation as:
5(2^(3x)) = 8
ln(5(2^(3x))) = ln(8)

Step 2: Use the properties of logarithms to simplify the equation:
ln(5) + ln(2^(3x)) = ln(8)

Step 3: Apply the power rule of logarithms to simplify further:
ln(5) + 3x ln(2) = ln(8)

Step 4: Rearrange the equation to isolate x:
3x ln(2) = ln(8) - ln(5)

Step 5: Combine the logarithmic terms using the quotient rule of logarithms:
3x ln(2) = ln(8/5)

Step 6: Solve for x by dividing both sides of the equation by 3 ln(2):
x = ln(8/5) / (3 ln(2))

Now, let's calculate the value of the expression ln(8/5) / (3 ln(2)) to compare it with your answer:
x ≈ ln(8/5) / (3 ln(2))
x ≈ 0.225 / (3 × 0.693)
x ≈ 0.225 / 2.08
x ≈ 0.108

As you can see, the value of x obtained from ln(8/5) / (3 ln(2)) is approximately 0.108, which differs from your provided answer of ln(1.6) / (3 ln(2)). Please double-check your calculations or provide further context if necessary.