a. To evaluate log base 3 of 18 using the change-of-base formula, we will use log base 10 and log base 3.
First, we can write the change-of-base formula as:
log base a of x = log base b of x / log base b of a
In this case, a=3 and x=18, so we can rewrite the logarithm as:
log base 3 of 18 = log base 10 of 18 / log base 10 of 3
Now, we can evaluate the logarithms using a scientific calculator:
log base 10 of 18 ≈ 1.255273
log base 10 of 3 ≈ 0.477121
Plugging these values back into the formula:
log base 3 of 18 ≈ 1.255273 / 0.477121 ≈ 2.630929
Therefore, log base 3 of 18 is approximately 2.630929, correct to six decimal places.
b. To evaluate log base 18 of 9 using the change-of-base formula, we will use log base 10 and log base 18.
Using the same change-of-base formula as before:
log base a of x = log base b of x / log base b of a
In this case, a=18 and x=9, so we can rewrite the logarithm as:
log base 18 of 9 = log base 10 of 9 / log base 10 of 18
Using a scientific calculator again:
log base 10 of 9 ≈ 0.954243
log base 10 of 18 ≈ 1.255273
Plugging these values back into the formula:
log base 18 of 9 ≈ 0.954243 / 1.255273 ≈ 0.759836
Therefore, log base 18 of 9 is approximately 0.759836, correct to six decimal places.
Use the change-of-base formula to evaluate each logarithm correct to six decimal places.
a. log base 3 of 18
b. log base 18 of 9
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