### Video Transcript

If the function π of π₯ is equal
to two raised to the power π₯ and the function π of ππ₯ equals five raised to the
power π₯, find the value of π, and state what type of transformation of the graph
π¦ equals π of π₯ this represents.

Weβre given two functions π of π₯
equals two raised to the power π₯ and π of ππ₯ equals five raised to the power
π₯. And we want to know what value of
π do we multiply π₯ by so that two raised to the power ππ₯ is equal to five raised
to the power π₯.

Now these are both exponential
functions, and we can use the laws of logarithms to simplify this type of
function. In particular, we use the power law
for logarithms, which tells us that the log to the base π of π raised to the power
π is equal to π times the log to the base π of π. That is, we bring the power or
exponent down to the front and multiply by it.

So if we take the logarithm on both
sides of our equation, say, to the base 10, where by convention we donβt write in
the base, we have log of two raised to the power ππ₯ is equal to the log of five
raised to the power π₯. Now using the power law for logs,
we can bring both our exponents down to the front and multiply by them. So we have ππ₯ times log two is
equal to π₯ times log five. And now, provided π₯ is nonzero, we
can divide through by π₯ log two. Dividing top and bottom on the
right by π₯ then leaves us with π equal to log five over log two. So weβve found our value for
π. Thatβs log five over log two.

Making a note of this and making
some space, we see that weβre also asked to state what type of transformation of the
graph π¦ equals π of π₯ this represents. To answer this, we recall that for
a constant π greater than zero, replacing π₯ with π times π₯ corresponds to a
horizontal scaling of the graph π¦ equals π of π₯. What this means is that π₯-values
of the graph of π¦ equals π of π₯ are divided by the constant π. In our case then, with π equal to
log five over log two corresponding to the constant π, the transformation of the
graph π¦ equals π of π₯ is a horizontal scaling.

Itβs worth pointing out that a
vertical scaling, as opposed to a horizontal scaling, of π¦ equals π of π₯ occurs
when each of the π¦-values is multiplied by a constant π. Itβs also worth noting that in the
case of a horizontal scaling of the graph π¦ equals π of π₯, a value of the
constant π greater than one represents horizontal shrinkage or compression of the
graph, whereas if π is between zero and one, the transformation is a horizontal
stretch of the graph π¦ equals π of π₯. Now in our case, with the constant
log five over log two, this is greater than one. So this corresponds more precisely
to a horizontal shrinkage or compression of the the graph π¦ equals π of π₯.

Finally, itβs important to
understand the distinction between algebraic scaling of the variable in the
expression, thatβs π of ππ₯, and what happens to the graph under this
transformation. In fact, the graph π¦ equals π of
π₯ is scaled by a factor of one over the constant π. And to reiterate, if π of π₯ is
two to the power π₯ and π of ππ₯ is five to the power π₯, then π is equal to log
five over log two. And this represents a horizontal
scaling of the graph π¦ equals π of π₯.