### Video Transcript

Solve two multiplied by three to
the power of π₯ is equal to five multiplied by four to the power of π₯ for π₯,
giving your answer to three decimal places.

In order to calculate an exponent
or power in an exponential equation, we need to use logarithms. There are a couple of ways of
approaching this problem. One way would be to get all the
terms within exponents of π₯ on one side of the equation. We can do this by dividing both
sides of the equation by two multiplied by four to the power of π₯. On the left-hand side, the twos
cancel leaving us with three to the power of π₯ over four to the power of π₯. On the right-hand side, weβre left
with five over two or five-halves.

We recall that the quotient of two
constants raised to the same exponent can be rewritten as shown. π to the power of π₯ divided by π
to the power of π₯ is equal to π over π all to the power of π₯. This means we can rewrite the
left-hand side of our equation as three-quarters to the power of π₯. We then recall the link between
exponentials and logarithms. If π to the power of π₯ is equal
to π, then π₯ is equal to log to the base π of π. As π is equal to three-quarters
and π is equal to five over two, then π₯ is equal to log to the base three-quarters
of five over two.

Typing this into the calculator
gives us negative 3.185081 and so on. As we want our answer to three
decimal places, the deciding number is the zero. As this is less than five, we round
down. Our value of π₯ correct to three
decimal places is negative 3.185.

We couldβve used an alternative
method from the line three-quarters to the power of π₯ is equal to five over
two. We could take a log of both sides
of the equation, such that log of three-quarters to the power of π₯ is equal to log
of five over two.

One of our laws of logarithms
states that log π to the power of π₯ is equal to π₯ log π. The left-hand side can be rewritten
as π₯ multiplied by log of three-quarters. We can then divide both sides of
the equation by log of three-quarters. π₯ is equal to log of five over two
divided by log of three-quarters. Once again, this gives us an answer
of negative 3.185081 which we know rounds to negative 3.185. We could check this answer by
substituting our value back into both sides of the original equation.