HSC Maths Advanced Syllabus

Logarithms and Geometric Sequences

Logarithms are frequently used in finance, particularly in conjunction with geometric sequences, to solve problems involving compound interest, growth rates, and the value of money. Here is an in-depth explanation of how solve logarithmic and exponential inequalities:

The first method is to simply use trial and error to solve the inequalities. This generally works when small values are used. For example, for the inequality $2^n<70$, we can simply substitute small values of $n$, to see what values result in values less than $70$.

Thus, we can substitute in $n=6$, as a rough estimate:

$$2^6 <70\\64<70$$

Therefore, we can see that $n=6$ fits the inequality as $64$ is less than $70$ . However, we also know that it is the largest integer that fits the inequality. It should be noted that as numbers grow in value, the accuracy of this method is diminished.

The second method is to turn the inequality into an equation, solve for the unknown variable, and then depending on the original inequality sign, determine the closest integer which fits the inequality. Let’s complete these steps using the example, $2^n<500000$.

Turning the inequality into an equation we get:

$$2^n=500000$$

Now we solve the equation to find $n$:

$$n=log_2500000\\=\frac{log500000}{log2}\\=18.93$$

Looking at the original inequality, we know that $2^n< 500000$, and as such the largest possible integer $n$ can equal to fit this inequality, is $18$.