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# Solving exponential equations using logarithms: base-10

Sal solves the equation 10^(2t-3)=7. Created by Sal Khan.

Video transcript

Voiceover:Solve the
equation for T and express your answer in terms
of base 10 logarithms. And this equation is 10 to
the 2T - 3 is equal to 7. We want to solve for T in
terms of base 10 logarithms. So let me get my little
scratchpad out and I've copied and pasted the same problem. So I'm just going to
rewrite it, so they have 10 to the 2T-3 is equal to 7. Actually, let me color
code this a little bit. So 10 to the 2T - 3 is equal to 7. So this is clearly an
exponential form right over here. if we want to write it
in logarithmic form, where we could, that'll
essentially allow us to solve for the exponent, so we could say, this is the exact same
truth about the universe as saying that the log base
10 of 7 is equal to 2T - 3. Let's just make sure that makes sense, this is saying 10 to the 2T - 3 = 7 This is saying that the power
that I need to raise 10 to to get to 7 is 2T - 3, or 10
to 2T - 3 power is equal to 7. So these are equivalent statements. What this form does is it
starts to put it into a form that's easier to solve for
T, now if we want to solve for T, we can add 3 to both sides. So if we add 3 to both
sides, we are going to get, log base 10 of 7 + 3, plus 3. And this +3 is of course
outside of the logarithm to make it clear, it's just like that. And on the right hand side
this is going to be equal to, this is going to be equal to just 2T. And now to solve for T we
just divide both sides by 2. So if we divide both sides by 2 we get, T is equal to all of this
business, log base 10 of 7 + 3 all of that over 2, so let me
see if I can remember this, and write it in the actual
box that they gave us. So we want to do log base
10 of 7 and then that + 3 yup it's formatting it
right, divided by 2. So that's what I'm
claiming the T is equal to in terms of Base 10 logarithms. So let me check my answer
and I got it right.