Further Logarithmic Equations

Video lesson · from 28:484 worked examples

One lesson video covers all of Exponentials & Logarithms, so it opens at 28:48 for this topic — not from the beginning.

Theory

Many mathematical models of real-life situations use exponential and logarithmic functions.

We have previously studied basic logarithmic equations and can now solve equations using the properties of logarithmic functions we have now become familiar with.

⚠️ Common Examiner Traps

Condense to a single log, then switch form: combine everything into one logarithm, then rewrite it as an exponential equation to solve.
Check every solution in the original: the algebra can produce values that make a logarithm undefined. Those must be rejected explicitly, not silently dropped.
Keep the lines consistent: conversions between log and exponential form are where working most often stops following logically.
A number can be written as a log: $2 = \log_a a^2$ , which lets you combine a constant with the log terms.

Worked examples

Example 1

Solve $\log_3 13 + \log_3 x = \log_3 273$ for $x>0$ .

\log_3(13x) = \log_3 273

\begin{aligned} 13x &= 273 \\ x &= \frac{273}{13} \\ x &= 21 \end{aligned}

Example 2

Solve $\log_3(4x+3) - \log_3(2x-3) = 1$ for $x > \frac{3}{2}$ .

\log_3\left(\frac{4x+3}{2x-3}\right) = 1

Convert to exponential form (base 3 to power 1):

\begin{aligned} \frac{4x+3}{2x-3} &= 3^1 \\ 4x+3 &= 3(2x-3) \\ 4x+3 &= 6x - 9 \\ -2x &= -12 \\ x &= 6 \end{aligned}

Example 3

Solve $\log_a(2p+1) + \log_a(3p-10) = \log_a(11p)$ for $p>4$ .

\log_a((2p+1)(3p-10)) = \log_a(11p)

\begin{aligned} (2p+1)(3p-10) &= 11p \\ 6p^2 - 20p + 3p - 10 &= 11p \\ 6p^2 - 17p - 10 &= 11p \\ 6p^2 - 28p - 10 &= 0 \\ \end{aligned}

Divide by 2:

\begin{aligned} 3p^2 - 14p - 5 &= 0 \\ (3p+1)(p-5) &= 0 \end{aligned}

So $p = -\frac{1}{3}$ or $p = 5$ . Since we are given $p > 4$ , then p = 5.

Example 4

Solve $\log_2 7 = \log_2 x + 3$ for $x>0$ .

Rewrite 3 as a logarithm base 2:

3 = 3\log_2 2 = \log_2(2^3) = \log_2 8

\begin{aligned} \log_2 7 &= \log_2 x + \log_2 8 \\ \log_2 7 &= \log_2(8x) \\ 7 &= 8x \\ x &= \frac{7}{8} \end{aligned}