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# Solving exponential equations using logarithms: base-10

## Video transcript

solve the equation 40 and express your answer in terms of base 10 logarithms and this equation says 10 10 to the 2t minus 3 is equal to 7 we want to solve for T in terms of base 10 logarithms so let me get my little scratch pad 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 minus 3 is equal to 7 actually let me color code this a little bit so 10 to the 2t 2 t minus 3 is equal to 7 so this is clearly an exponential form right over here if we want to write it in logarithmic form weird that will 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 log base 10 log base 10 of 7 is equal to is equal to 2t minus 3 2 t minus 3 let's just make sure that makes sense this is saying 10 to the 2 3 minus 3 is equal to 7 this is saying that the power that I need to raise 10 to to get to 7 is 2t minus 3 or 10 to the 2t minus 3 power is equal to 7 so these are equivalent statements what this form does is it starts to put 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 add 3 to both sides we are going to get log base 10 of 7 of 7 plus 3 plus plus 3 and this plus 3 is of course outside of the logarithm to make it clear it's like well 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 Avot log log base 10 of 7 plus 3 all of that over 2 so let me see if I can if I can remember this and write it in my little light and write and write it in the actual box that they gave us so we want to do log base 10 of 7 and then that plus 3 yep it's formatting it right / - so that's what I'm claiming that T is equal to in terms of base 10 logarithms so let me check my answer and I got it right