Question

Solve this equation for x. Round your answer to the nearest hundredth.

0.77 = log x

Answer 1

This is a common logarithm, or a logarithm base 10.
\[0.77 = \log_{10} x\]The meaning of a logarithm \[a = \log_b x\]is that \(b^a=x\)

That means we have \[10^{0.77} = x\]

Answer 2

so 10^0.77=x is the answer

Answer 3

Well, no, because the problem asks you to round the answer to the nearest hundredth. They want you to evaluate \(x = 10^{0.77}\). I can tell you that it is approximately 6, but that isn't close enough to get the problem correct.

We started with \[0.77 = \log_{10} x\]One property of logs is that \[\log ab = \log a + \log b\]I happen to know that \[\log_{10} 2 \approx 0.30103\]and\[\log_{10}3 \approx 0.47712\]Notice that adding those two numbers together gives me 0.77815, which is awfully close to 0.77. That means 2*3 is just slightly larger than the number whose common log is 0.77.

By memorizing a handful of logs of prime numbers (and remembering the properties of exponents and logs), you can work out good estimates of many problems without a calculator. Learn a few yourself and impress your friends :-)