Laws of logarithms
evualuat using change of base rule
log(√8)16
^the square root 8 is supposed to be subscript.
the answer i got was 8/3, not sure if that is right
You are correct
This can be done without a calculator
let log √816 = x
(√8)^x = 16
( (2^3)^(1/2)^x = 2^4
3(1/2)x = 4
3x = 8
x = 8/3
Thanks very much Reiny :)
welcome
To evaluate log(√8)16 using the change of base rule, you need to apply the following steps:
Step 1: Identify the base of the logarithm you are working with. In this case, it's not specified, so let's assume it is base 10.
Step 2: Apply the change of base rule, which states that log base a of b can be written as log base c of b divided by log base c of a.
In this case, you are evaluating log(√8)16, so you need to choose a common base for the logarithm. Let's choose base 10 and rewrite the expression using the change of base rule as follows:
log(√8)16 = log10 16 / log10 (√8)
Step 3: Simplify the expression.
The logarithm of 16 to base 10 (log10 16) can be easily calculated as log(16) / log(10). Using a calculator, log(16) is approximately 1.2041 and log(10) is 1.
To evaluate the logarithm of √8 to base 10 (log10 (√8)), we can write it as log10 (8^(1/2)). By applying the power rule of logarithms, this can be simplified as (1/2) * log10 8.
The logarithm of 8 to base 10 (log10 8) can be calculated as log(8) / log(10). Using a calculator, log(8) is approximately 0.9031.
Now, substitute the obtained values into the expression:
log(√8)16 = log10 16 / log10 (√8)
= 1.2041 / (0.5 * 0.9031)
Perform the calculations:
= 1.2041 / 0.4516
≈ 2.6667
Please note that the answer is approximately 2.6667, not 8/3.