Why do logarithms turn multiplication into addition

Exponentiation and logarithms are opposite operations. When you raise a base (a) to a sum of exponents, the rule (a^{b+c}=$a^b$\cdot $a^c$) shows that addition in the exponent becomes multiplication of the results. The logarithm, being the inverse of this process, reverses the direction: a product inside the log becomes a sum of logs.

Why multiplication becomes addition

Start with the definition (y=\log_a x\) which means ($a^y$ = x). If (x) is a product (x=uv), then (a^{\log_a u}$\cdot$ a^{\log_a v}=uv). Using the exponent rule (a^{p}$\cdot$ a^{q}=a^{p+q}), we get (a^{\log_a u+\log_a v}=uv). Since the exponential function is one‑to‑one, the exponents must be equal, giving $\log_a(uv)=\log_a u+\log_a v$. This is the product‑to‑sum property.

Key exponent and log rules that underlie the property:

• (a^{b+c}=$a^b$ $a^c$\)
• ( (ab)^c = $a^c$ $b^c$\)
• $\log_a(xy)=\log_a x+\log_a y$

Applying the product rule in a calculation:

1. 1Write each factor as a power of the same base.
2. 2Take the logarithm of the product.
3. 3Replace each factor with its logarithm and add them.

Sample values of $\log_2$ and how they add:

A concrete example solidifies the idea. Compute $\log_2(8\times4)$. Using the product rule, $\log_2 8 + \log_2 4 = 3 + 2 = 5$. Directly, (8\times4=32) and $\log_2 32 = 5$, confirming that multiplication inside the log became addition outside. Remember to keep the base consistent, and the property works for any number of factors.