Using logarithms, find w to the nearest hundredth: 5^(2w) +9=40?
5^(2w) = 31
2w log 5 = log 31
w = (1/2) (log 31/log 5)
To solve the equation 5^(2w) + 9 = 40 using logarithms, follow these steps:
Step 1: Subtract 9 from both sides of the equation:
5^(2w) = 40 - 9
5^(2w) = 31
Step 2: Take the logarithm of both sides of the equation. You can choose any base for the logarithm, but commonly used bases are 10 (log) and e (ln):
log(5^(2w)) = log(31)
Step 3: Apply the power rule of logarithms by bringing down the exponent 2w as a coefficient:
(2w)log(5) = log(31)
Step 4: Divide both sides of the equation by log(5) to isolate 2w:
2w = log(31) / log(5)
Step 5: Now divide the denominator log(5) into the numerator log(31) using a calculator to evaluate:
2w ≈ 1.4967 / 0.69897
Step 6: Perform the division to find the value of 2w:
2w ≈ 2.1412
Step 7: Finally, divide both sides of the equation by 2 to solve for w:
w ≈ 2.1412 / 2
Therefore, w ≈ 1.07 (rounded to the nearest hundredth).
To solve the equation 5^(2w) + 9 = 40 using logarithms, we can first isolate the exponent term by subtracting 9 from both sides:
5^(2w) = 40 - 9
5^(2w) = 31
Next, we can take the logarithm of both sides with respect to the base 5. The logarithm function with base 5 is denoted as log5.
log5(5^(2w)) = log5(31)
Using the logarithm property logb(b^x) = x:
2w = log5(31)
Now, to solve for w, we divide both sides of the equation by 2:
w = (1/2) * log5(31)
Using a scientific calculator or any calculator with logarithmic functions, we can evaluate log5(31) and then multiply it by (1/2). Finally, we round the result to the nearest hundredth.
Therefore, w is approximately equal to the nearest hundredth.