Question - Solving Exponential Equation with Logarithms
Solution:
The equation given in the image is:5^(3-x) + 2 = 8To solve for x, we should first isolate the exponential term on one side of the equation:5^(3-x) + 2 - 2 = 8 - 25^(3-x) = 6Now, to solve for x, we need to rewrite the equation in logarithmic form. Taking the logarithm of both sides (using a common logarithm, base 10, or a natural logarithm, base e), we can apply the properties of logarithms to solve for the exponent. But first, let's use the natural logarithm, as it's often the one used in these types of problems:ln(5^(3-x)) = ln(6)Applying the property of logarithms that allows us to bring down the exponent, we get:(3 - x)ln(5) = ln(6)Now, divide both sides by ln(5) to isolate (3 - x):3 - x = ln(6) / ln(5)Now, solve for x:x = 3 - ln(6) / ln(5)Using a calculator to find the values of ln(6) and ln(5) and then performing the subtraction:x ≈ 3 - (1.79176 / 1.60944)x ≈ 3 - 1.1134x ≈ 1.8866Rounded to the nearest thousandth, the value of x is approximately:x ≈ 1.887
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