Using Logs to Solve Equations with Powers of
You might be asked to solve an equation with an unknown power (e.g. ). For this example you could probably use trial and error and work out that . However the equation may not be so straightforward, in which case you would use logs to solve it.
You will need to remember the 3rd rule of logs discussed in the last lesson:
This rule will bring down from the power so that you can solve the equation.
Examples:
- Solve the equation:
Step 1: Take the log of both sides:
Step 2: use the rule to bring the power to the front:
Step 3: Tidy up and solve the equation as normal:
(N.b.Engineers often use the natural log instead of log, the method is exactly the same. Try the above example using and check that you get the same answer.
- Solve the equation:
Step 1:Take logs of both sides:
Step 2:Bring the power to the front:
Step 3:Tidy up and solve as normal:
- Solve the equation:
This question needs an extra step. You need to remember the rules of indices that states that if you multiply numbers with the same base then you add the powers.
Step 0: Using the rules of indices (add the powers):
Step 1: Take logs of both sides:
Step 2: Bring the power to the front:
Step 3:Tidy up and solve as normal:
Remember that for all of the above equations you can check your answer by substituting it back in the original equation to see if it works.
For example:
(Try this on your calculator and you’ll see that the value of is correct.)
- Find and when: and
When asked to solve equations with 2 unknown values (in this case and ) we will consider each equation separately at 1st and once we have moved the unknowns from the power (using logs) we will solve them simultaneously.
Equation 1:
Step 1: Take logs of both sides:
Step 2: Bring the power to the front:
Step 3: Tidy up:
Equation 2:
Step 1: Take logs of both sides:
Step 2: Bring the power to the front:
Step 3: Tidy up:
Now we have 2 equations that can be solved simultaneously (see lecture 4 for more information on solving simultaneous equations):
Add the equations to get:
Substitute in equation 1:
We have found that the values of and will work in both equations. Use your calculator to check that the values are correct.
An alternative method:
If you are happy with the above method for solving this type of equation you don’t need to read the following. It’s just included for completeness as some people like to think about it this way.
Remember what a log means:
e.g.means“what power do I need to raise 3 to, to get 3?” (answer=1)
So if the base and the number are the same then the answer will always be 1.
In general:
We could use this fact to solve this type of equation.
- Solve the equation:
Step 1: Take the log of both sides using 4 as the base:
Step 2: use the log rule to bring the power to the front:
Step 3: we know that so we now have:
Step 4: Tidy up and solve the equation as normal.
© H Jackson 2011 / ACADEMIC SKILLS1