Solve giving answer to 4 s.f

Question

2^x × 3^x = 5^(x + 1)

Answer 1

To solve the equation 2^x × 3^x = 5^(x + 1), we can use logarithms.

Taking the logarithm (base 10) of both sides of the equation, we get:
log(2^x × 3^x) = log(5^(x + 1))

Using the logarithmic properties, we can simplify this to:
x(log2 + log3) = (x + 1)log5

Expanding and rearranging, we have:
x log2 + x log3 = x log5 + log5

Now, we can solve for x by isolating it on one side of the equation:
x log2 - x log5 = log5 - x log3
x(log2 - log5) = log5(1 - log3)

Dividing both sides by (log2 - log5), we get:
x = log5(1 - log3) / (log2 - log5)

To evaluate this expression to 4 significant figures, we can use the following approximate values:
log2 ≈ 0.3010
log3 ≈ 0.4771
log5 ≈ 0.6989

Substituting these values into the equation, we have:
x ≈ 0.6989(1 - 0.4771) / (0.3010 - 0.6989)
x ≈ 0.6989(0.5229) / (-0.3979)
x ≈ -0.1827 / (-0.3979)
x ≈ 0.4593

Therefore, the solution to the equation, rounded to 4 significant figures, is x ≈ 0.4593.