Question - Solving Exponential Equation with Logarithms
Solution:
The equation provided in the image is:5^(3x) + 2 = 8To solve for x, you should follow these steps:1. Subtract 2 from both sides to isolate the exponential term: 5^(3x) + 2 - 2 = 8 - 2 5^(3x) = 62. Take the logarithm of both sides of the equation. It's often easier to use the logarithm with the same base as the exponent, but you can use the natural logarithm (ln) or common logarithm (log base 10): ln(5^(3x)) = ln(6)3. Use the power rule of logarithms, which states that ln(a^b) = b*ln(a): 3x * ln(5) = ln(6)4. Divide both sides by 3*ln(5) to solve for x: x = ln(6) / (3 * ln(5))Now, we can compute the value of x numerically. Note that since you're asked to round to the nearest thousandth, you should use a calculator to ensure accuracy. x ≈ ln(6) / (3 * ln(5))Using a calculator: x ≈ ln(6) / (3 * ln(5)) x ≈ 1.791759469 / (3 * 1.609437912) x ≈ 1.791759469 / 4.828313737 x ≈ 0.371072Rounding to the nearest thousandth: x ≈ 0.371So, x is approximately 0.371 when rounded to the nearest thousandth.
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