Logarithms (VCE SSCE General Mathematics): Revision Notes

Worked Example: Evaluating log(100)

Let's find the logarithm of 100.

Step 1: Write 100 as a power of 10

$100 = 10^2$

Step 2: Write down the logarithm

$\log(100) = \log(10^2) = 2$

Answer: $\log(100) = 2$ ✓

Worked Example: Finding log(45)

Find $\log(45)$ to one decimal place.

Step 1: Enter log(45) on your calculator

Using a CAS calculator, enter log(45) and press ENTER or EXE.

The calculator displays: $\log_{10}(45) = 1.65321$

Step 2: Round to one decimal place

$\log(45) = 1.65... = 1.7$ (to one decimal place)

Answer: $\log(45) = 1.7$ ✓

Worked Example: Finding the number when log is known

Find the number whose logarithm is 3.1876 to one decimal place.

Step 1: Identify the relationship

If $\log(x) = 3.1876$, then $x = 10^{3.1876}$

Step 2: Calculate using a calculator

Enter $10^{3.1876}$ on your calculator.

Result: $10^{3.1876} = 1540.281...$

Step 3: Round to one decimal place

$x = 1540.3$ (to one decimal place)

Answer: The number is 1540.3 ✓

Worked Example: Order of magnitude

What is the order of magnitude of 1200?

Step 1: Write 1200 in standard form

$1200 = 1.2 \times 10^3$

Step 2: Identify the power of 10

The power of 10 is 3, so the order of magnitude is 3.

Note: The order of magnitude of 1.2 itself is 0, since $1.2 = 1.2 \times 10^0$

Answer: The order of magnitude of 1200 is 3 ✓

Worked Example: Transforming data using log₁₀(x)

For the table below, apply a $\log_{10}(x)$ transformation to check if this creates a straight-line relationship.

Step 1: Calculate $\log_{10}(x)$ for all $x$ values

Step 2: Plot $y$ against $\log_{10}(x)$

Remember to label the horizontal axis as $\log_{10}x$.

Step 3: Check for linearity

The graph shows a clear straight-line relationship. The data has been successfully linearised using the $\log_{10}(x)$ transformation.

Answer: The $\log_{10}(x)$ transformation has linearised the data. ✓

Worked Example: Calculator log transformation

For the table below, use a CAS calculator to apply the $\log_{10}(x)$ transformation.

Steps for TI-Nspire CAS:

1. Start a new document and select "Add Lists & Spreadsheet"
2. Enter the data into columns named $x$ and $y$
3. Create a third column named "logarx"
4. In the cell below "logarx", enter the formula =log10(x) and press ENTER
5. Create a scatterplot of $y$ against $\log_{10}x$ using "Add Data & Statistics"

Steps for CASIO ClassPad:

1. In the Statistics application, enter data into columns named $x$ and $y$
2. Create a third column named "log x"
3. In the calculation cell at the bottom of the third column, type log10(x)
4. Create a scatterplot by setting XList to "main\logx" and YList to "main\y"

The resulting scatterplot will show whether the transformation has linearised the data.

Worked Example: Mammal body weight and heart rate

Let's explore the relationship between mammal body weight and heart rate.

The data:

Challenge: The body weights range from 2.5 grams (shrew) to 170,000,000 grams (blue whale). This enormous range makes it difficult to plot on a normal scale.

Solution: Use logarithmic transformation.

Step 1: Calculate $\log(\text{weight})$ for each mammal

Step 2: Plot heart rate against $\log(\text{body weight})$

Observation: The logarithmic transformation allows us to visualise the relationship clearly. Smaller animals (low log weight values) have faster heart rates, while larger animals (high log weight values) have slower heart rates.

Worked Example: Comparing animal weights

Using the logarithmic values from the table above, how many times heavier is a giraffe than a rabbit?

Step 1: Subtract the logarithms

$\log(\text{giraffe weight}) - \log(\text{rabbit weight}) = 5.95 - 3.00 = 2.95$

Step 2: Interpret the difference

The difference of 2.95 means the giraffe is $10^{2.95}$ times heavier than the rabbit.

Step 3: Calculate $10^{2.95}$

$10^{2.95} = 891.25$

Step 4: Round to nearest hundred

$891.25 \approx 900$

Answer: A giraffe is approximately 900 times heavier than a rabbit. ✓