LOGARITHMS AND EQUATION SOLVING
Solve for x. Round to the nearest hundredth.
2^(x+3) = 17^(x)
take the ln of both sides
(x+3)*ln2=xln17
x(ln2-ln17)=-3ln2 and solve for x.
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To solve the equation 2^(x+3) = 17^(x), we can use logarithms. Here's the step-by-step explanation of how to solve this equation:
Step 1: Take the logarithm of both sides of the equation. You can use any logarithm base you prefer, but commonly used bases are 10 (log) or e (ln).
Using natural logarithm (ln) for this example, we have:
ln(2^(x+3)) = ln(17^x)
Step 2: Apply the logarithmic property to bring the exponent down:
(x+3) ln(2) = x ln(17)
Step 3: Expand by distributing the logarithmic values:
x ln(2) + 3 ln(2) = x ln(17)
Step 4: Gather like terms by subtracting x ln(2) from both sides:
3 ln(2) = x ln(17) - x ln(2)
Step 5: Factor out the common factor, which is x, from the right side:
3 ln(2) = x (ln(17) - ln(2))
Step 6: Divide both sides of the equation by (ln(17) - ln(2)) to solve for x:
x = (3 ln(2)) / (ln(17) - ln(2))
Step 7: Calculate the value of x using a calculator and rounding it to the nearest hundredth.
Note: Make sure you use parentheses around ln(2) and ln(17) when typing the expression into the calculator.
By following these steps and performing the necessary calculations, you'll find the approximate value of x.