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Logarithms is the inverse function of exponents, analogous to how subtraction is the opposite of addition and multiplication is the opposite of division.

Specifically, if we have $a^b=c$, then we also have $\log_ac=b$.

If you see $\log x$ without a base, then it is automatically assumed to be in base $10$. In other words, $\log x=\log_{10}x$.

Similarly, $\ln x$ is the logarithm base $e$. So $\ln x=\log_ex$.

To illustrate the fact that $\log_a x$ is indeed the inverse of $a^x$, we will compare the graphs of $f(x)=e^x$ and $g(x)=\ln x$:

As you can see, the graph of $f(x)=e^x$ and $g(x)=\ln x$ are mirror images of each other, flipped over the line $y=x$; a characteristic property of inverse functions.

Now let's look at some of the properties of logarithms:

Logarithm Sum Identity: $\log_a xy=\log_a x + \log_a y$

Logarithm Difference Property: $\log_a \dfrac{x}{y} = \log_a x-\log_a y$

Logarithm Division Property: $\dfrac{\log_a x}{\log_a y}=\log_y x$

Logarithm Power Property: $\log_a x^n = n\log_a x$

Logarithm Reciprocal Property: $\log_a b = \dfrac{1}{\log_b a}$

Logarithm Chain Rule: $(\log_b a)(\log_a c) = \log_b c$

You can try to prove these rules by converting them into more familiar exponents. For example, we can prove the first property as follows:

Let $\log_a x=m$ and $\log_a y=n$. Thus, $a^m=x$ and $a^n=y$.

Multiplying the equations together, we obtain $a^{m+n}=xy$.

Converting back into logarithm form, we have $\log_a xy=m+n=\log_a x+\log_a y$, as > desired. $\Box$