Solving equations using logs

You may not be able to find log₂ 10000000 on the calculator, a calculator may have log₁₀ and not log₂. So instead we take logs on both sides to base 10 we use;

So now we have;

…now we divide both sides by log 2 to get;

Now it is easier to use the calculator to work it out.

When worked out we see that;

…round it;

We can check it in the original formula to see whether the answer is correct.

…again we need to use;

…to get;

…simplify…

…we use logs to get;

…we use;

to get;

We have to reverse the inequality sign because log 0.3 is negative.

We can conclude that the smallest integer must be 8.

Normally to find the intersection we need to put the equations equal so we have.

…use logs to solve the equations

…we rearrange the equation using;

…to get;

…we divide through by 3

…we work out the log division

…now we simplify.

The curves intersect at x = -0.613

Now we can sketch the graph, first we sketch

The graph is simply y=2^x reflected in the y axis.
[graph id=”26″]
Next we sketch 3^{x + 1}
[graph id=”27″]
On the graph we can see that the curves intersect at;

Now we can use;

…to get;

…we use the subtraction rule to get;

By doing a power 3 on the opposite side we can remove log³

…we simplify to get;

…then we rearrange to get;

Now we can factorise and solve;