Video Transcript
Solve 17 to the power of 𝑥 is
equal to two for 𝑥, giving your answer to three decimal places.
We’re given an equation, and we
need to solve this equation for 𝑥. And we need to give our answer to
three decimal places. To do this, let’s start by looking
at our equation. We can see that 𝑥 is the exponent
or power of this equation. This should remind us of
logarithms, because remember, logarithmic functions are the inverse of exponential
functions. And in fact, this gives us two
different ways of solving this equation.
The first way is to recall exactly
what we mean when we say that logarithmic functions are inverse of exponential
functions. This means if 𝑎 to the power of 𝑥
is equal to 𝑏, this means that 𝑥 must be equal to the logarithm base 𝑎 of 𝑏. And one way of thinking about this
is to take the logarithm base 𝑎 of both sides of our original equation. On the right-hand side, we would
get the log base 𝑎 of 𝑏. And on the left-hand side, we would
have the log base 𝑎 of 𝑎 to the power of 𝑥.
And then, we use the fact that
these are inverse functions to just get 𝑥. We want to apply this to the
equation given to us in the question: 17 to the power of 𝑥 is equal to two. In this case, our value of 𝑎 is 17
and our value of 𝑏 is two. So, we must have that 𝑥 is equal
to the logarithm base 17 of two. And if we put this into our
calculator, we get 0.2446 and this continues.
But remember, the question only
wants us to give this to three decimal places. Since the fourth decimal place is
six, we’re going to need to round up. And by doing this, we get, to three
decimal places, 𝑥 is 0.245. But this is only one way we could
have solved this equation. Let’s now look through a second
method.
We’re going to want to take
logarithms of both sides of our original equation. And in fact, it won’t matter which
logarithm base we choose. So, we’ll take the logarithm base
10. Remember, we can represent this by
just writing log. So now, we have the log base 10 of
17 to the power of 𝑥 is equal to the log base 10 of two. And now, we can see we’re taking
the logarithm of an exponential function.
We need to recall the following
rule for logarithms, which is often called the power rule for logarithms. The log base 𝑎 of 𝑏 to the 𝑛th
power is equal to 𝑛 times the log base 𝑎 of 𝑏. Essentially, this means if we’re
taking the logarithm of an exponential function, we can instead multiply it by our
exponent. So by using this on the left-hand
side of our equation, we can rewrite the log base 10 of 17 to the power of 𝑥 as 𝑥
times the log base 10 of 17.
Finally, all we need to do is
rearrange this equation for 𝑥 by dividing both sides of our equation through by the
log base 10 of 17. This gives us that 𝑥 is equal to
the log base 10 of two divided by the log base 10 of 17. And if we calculate this, we get
exactly the same answer we did before. To three decimal places, 𝑥 is
0.245.
Therefore, we saw two different
methods for solving the equation 17 to the power of 𝑥 is equal to two. In both of these methods, to three
decimal places, 𝑥 was equal to 0.245.
