Without calculator

Let's start with the definition of a logarithm: if $a^x = b$, then $\log_a(b) = x$. Can anyone summarize what this means in terms of bases and exponents?

It means that the logarithm tells us the exponent we need to raise the base to in order to get to a certain number.

Exactly! Think of logarithms as the inverse operation to exponentiation. Can someone give me an example using this definition?

If $2^3 = 8$, then $\log_2(8) = 3$.

Great job! This initial understanding will help us evaluate logarithmic expressions.

Now, let's evaluate some logarithms without calculators. For instance, how would we evaluate $\log_8$?

Is it $log_8 = 3$ because $2^3 = 8$?

Exactly! Let's try another: how about $\log_{81}$?

That would be $4$ since $3^4 = 81$!

Perfect! Being comfortable with these calculations is important for our further explorations in logarithms.

Let's apply what we've learned in a scenario: if I ask you to find $\log_2(16)$, how would you solve it?

Since $2^4 = 16$, then $\log_2(16) = 4$.

Can I also express it as $\log_2(4^2)$ and use the power rule?

Absolutely! That's a great application of the power rule: $\log_a(m^k) = k \cdot \log_a(m)$. Well done!

This helps me remember: logarithm rules connect with exponents!