Topic 2.12 Notes – Logarithmic Function Manipulation

What Logarithm Manipulation Means

A logarithm answers the question “what exponent gives this value?” If $\log_b x=u$, then $b^u=x$. That is why logs turn exponent rules into log rules.

Because of that:

• multiplication inside a log becomes addition outside
• division becomes subtraction
• a power on the input becomes a coefficient

The restrictions never go away:

• Base restrictions for $\log_b x$ are $b>0$ and $b\ne 1$
• Argument restriction is $x>0$
• Natural log means base $e$, so $\ln x=\log_e x$, and it follows all the same rules

One warning shows up constantly on quizzes:

• $\log_b(x+y)\ne \log_b x+\log_b y$
• $(\log_b x)^n\ne n\log_b x$

Those are different structures. A sum inside one log does not split, and a power on the whole log output is not the power property.

The Log Properties You Use to Rewrite Expressions

$$\log_b(xy)=\log_b x+\log_b y$$

$$\log_b\left(\frac{x}{y}\right)=\log_b x-\log_b y$$

$$\log_b(x^n)=n\log_b x$$

$$\log_a x=\frac{\log_b x}{\log_b a}=\frac{\log x}{\log a}=\frac{\ln x}{\ln a}$$

A few exact values save time:

• $\log_b 1=0$
• $\log_b b=1$
• $\log_b(b^r)=r$

A few details matter:

• Product property works both ways. You can expand $\log_b(xy)$ or condense $\log_b x+\log_b y$.
• Quotient property also works both ways, but $\dfrac{\log_b x}{\log_b y}$ is just a quotient of two numbers, not a quotient-property rewrite.
• Power property reverses as coefficient $\to$ exponent. The coefficient must multiply the whole log.
• Change of base keeps the original base in the denominator. Students often flip it by mistake.

Expanding and Condensing Logarithms

Expanding

When you expand, peel the expression apart in this order:

1. products become sums
2. quotients become differences
3. powers move out front
4. simplify exact log values

Example:

$$\log_5\left(\frac{125x^4}{y^2}\right) =\log_5 125+\log_5(x^4)-\log_5(y^2) =3+4\log_5 x-2\log_5 y$$

Condensing

When you condense, reverse that flow:

1. move coefficients into exponents
2. combine sums into products
3. combine differences into quotients

Example:

$$3\log_2 p+\log_2 q-2\log_2 r =\log_2(p^3)+\log_2 q-\log_2(r^2) =\log_2\left(\frac{p^3q}{r^2}\right)$$

Only combine logs with the same base.

Domain habit

Keep the original restrictions even after rewriting. For

$$\log_b(x-4)+\log_b(x+1)$$

you need both $x-4>0$ and $x+1>0$, so the domain is $x>4$. After condensing, that restriction still stays.

How the Properties Show Up in Graphs

These rules are also graph facts.

• Horizontal scaling becomes a vertical shift

  $$\log_b(kx)=\log_b k+\log_b x$$

  So $f(kx)$ matches the graph of $f(x)=\log_b x$ shifted vertically by $\log_b k$.

• Power on the input becomes vertical scaling

  $$\log_b(x^k)=k\log_b x$$

  Replacing $x$ with $x^k$ multiplies all outputs by $k$.

• Changing the base changes vertical scale

  $$\log_a x=\frac{1}{\log_b a}\log_b x$$

  So all log graphs are vertical dilations of each other. They all have domain $x>0$, vertical asymptote $x=0$, and pass through $(1,0)$.

Common Mistakes and Fast Checks

• Splitting a sum inside a log, like $\log(x+y)$
• Combining logs with different bases
• Moving a coefficient into only part of the argument
• Writing change of base backward as $\dfrac{\log_b a}{\log_b x}$
• Forgetting domain restrictions after expanding or condensing
• Writing $\ln(x^2)=2\ln x$ without noting that this only works for $x>0$; for $x\ne 0$, the broader identity is $2\ln|x|$