Example Question - isolate exponential term
Here are examples of questions we've helped users solve.
Solving Exponential Equation with Logarithms
The equation given in the image is: 5^(3-x) + 2 = 8 To solve for x, we should first isolate the exponential term on one side of the equation: 5^(3-x) + 2 - 2 = 8 - 2 5^(3-x) = 6 Now, 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 - 1.1134 x ≈ 1.8866 Rounded to the nearest thousandth, the value of x is approximately: x ≈ 1.887
Solving Exponential Equations with Natural Logarithms
To solve the equation provided in the image, we should isolate the exponential term on one side: \( e^{4-7x} + 11 = 20 \) Subtract 11 from both sides: \( e^{4-7x} = 20 - 11 \) \( e^{4-7x} = 9 \) Now take the natural logarithm (ln) of both sides to remove the base e: \( \ln(e^{4-7x}) = \ln(9) \) By the property of logarithms, \(\ln(e^{y}) = y\), so: \( 4 - 7x = \ln(9) \) Now to isolate x, we first subtract 4 from both sides: \( -7x = \ln(9) - 4 \) Finally, we divide by -7: \( x = \frac{\ln(9) - 4}{-7} \) \( x = \frac{-4 - \ln(9)}{7} \) This is the solution for x in terms of natural logarithms.
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