Topic 2.9 Notes – Logarithmic Expressions

What a Logarithm Means

The definition is the center of everything:

$$\log_b c=a \iff b^a=c$$

That means:

• base $b$ is the number being raised to a power
• argument $c$ is the result you want to get
• value $a$ is the exponent

So $\log_4 64=3$ because $4^3=64$.

A few conditions matter for real logarithms:

• $b>0$
• $b\neq1$
• $c>0$

The result $a$ can be any real number:

• positive, like $\log_3 243=5$
• zero, like $\log_{12}1=0$
• negative, like $\log_5\left(\frac{1}{125}\right)=-3$
• fractional, like $\log_{16}8=\frac34$

If the base is left out, it means common logarithm, base 10:

$$\log c=\log_{10}c$$

Evaluating Logarithmic Expressions

Every evaluation problem becomes easier when you rewrite it as an exponential equation.

The basic move

If you see $\log_b c$, think:

$$b^a=c$$

and solve for $a$.

Anchor values

These show up constantly and are worth knowing cold:

$$\log_b 1=0$$

$$\log_b b=1$$

$$\log_b(b^a)=a$$

$$b^{\log_b c}=c$$

Common cases

• Positive exponent
  $\log_3 243=5$ because $3^5=243$
• Zero exponent
  $\log_{12}1=0$ because $12^0=1$
• Negative exponent
  $\log_5\left(\frac{1}{125}\right)=-3$ because $5^{-3}=\frac{1}{125}$
  $\log_7\left(\frac{1}{49}\right)=-2$ because $7^{-2}=\frac{1}{49}$
• Fractional exponent
  $\log_{16}8=\frac34$ because $16^{3/4}=8$
• Base between 0 and 1
  $\log_{\frac12}8=-3$ because $\left(\frac12\right)^{-3}=8$
• Common log
  $\log 0.001=-3$ because $10^{-3}=0.001$

A very common mistake is swapping the exponent and argument. In $\log_b c=a$, the answer is the exponent, not the argument.

Estimating and Checking Logarithm Values

Some logs are exact from exponent facts. Others are not, so you estimate with technology.

Before using a calculator, bound the answer with powers of the base. For example,

$$10^1<73<10^2$$

so

$$1<\log 73<2$$

Then a calculator gives:

$$\log 73\approx1.8633$$

That estimate makes sense because $73$ is between $10$ and $100$, so its common log should be between $1$ and $2$.

Sign checks help too:

• If base $>1$:
  • argument $>1$ gives a positive log
  • argument between $0$ and $1$ gives a negative log
• If base is between $0$ and $1$, those signs reverse.

Also, $\log 73$ is the exact value. $1.8633$ is only an approximation.

Reading Logarithmic Scales

A logarithmic scale uses equal spacing for equal multiplicative change, not equal difference.

On a base-10 log scale:

• moving right 1 unit means multiply by $10$
• moving left 1 unit means divide by $10$
• moving $k$ units means multiply by $10^k$

More generally, on a base-$b$ log scale, a change of 1 unit means multiply by $b$, and a change of $k$ units means multiply by $b^k$.

Examples you should recognize:

• $1,10,100,1000,10000$ are equally spaced on a base-10 log scale
• from $10^2$ to $10^5$ is 3 log units, so the original values differ by a factor of $10^3=1000$
• halfway between $10$ and $100$ is not $55$

That midpoint is $10^{1.5}\approx31.6$, because halfway on a log scale means halfway in the exponent.

Zero and negative original values do not appear on an ordinary log scale. Also, a log coordinate of $0$ means the original value is $1$, since $10^0=1$.

Common Mistakes and Quick Fixes

• Converting $\log_b c=a$ incorrectly. Keep the base as the base, the answer as the exponent, and the argument as the result.
• Treating $\log_b c$ like multiplication. It represents one number, the exponent.
• Forgetting $c>0$. $\log_b 0$ and $\log_b(-5)$ are undefined in the reals.
• Assuming logs must be positive. They can be negative or fractional.
• Forgetting that $\log x$ means base 10 in this course.
• Reading log-scale spacing as equal differences instead of equal ratios.
• Confusing log value $0$ with original value $0$.