Solve giving answer to 4 s.f
2^x × 3^x = 5^(x + 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.
To solve the equation 2^x × 3^x = 5^(x + 1), we need to apply logarithms.
Step 1: Take the logarithm of both sides of the equation. We can choose any base, but for simplicity, let's use the natural logarithm (base e). Applying ln (natural logarithm) to both sides, we get:
ln(2^x × 3^x) = ln(5^(x + 1))
Step 2: Use the logarithmic property that ln(a^b) = b * ln(a) to simplify the equation. Applying this property, we have:
x * ln(2) + x * ln(3) = (x + 1) * ln(5)
Step 3: Distribute the terms on the left side:
x * (ln(2) + ln(3)) = (x + 1) * ln(5)
Step 4: Expand the terms on both sides:
x * ln(6) = x * ln(5) + ln(5)
Step 5: Subtract x * ln(5) from both sides:
x * ln(6) - x * ln(5) = ln(5)
Step 6: Factor out x on the left side:
x * (ln(6) - ln(5)) = ln(5)
Step 7: Divide both sides by (ln(6) - ln(5)):
x = ln(5) / (ln(6) - ln(5))
Using a calculator for the calculation:
x ≈ 0.3348
Therefore, the approximate solution to the equation 2^x × 3^x = 5^(x + 1) to 4 significant figures is x ≈ 0.3348.