# Mth141s53 - Sec 5.3 Logarithms and Their Properties First...

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Sec. 5.3 – Logarithms and Their Properties First, let’s do a little preliminary investigation! In the last section we used the exponential formula : 1 nt r A P n = + Now in this formula, there are 5 variables. Right?? A, P, r, n, and t. Now if we are given any four of these values, we should be able to solve for the fifth one. Let’s see: we did the case where we knew P, r, n, and t and solved for A. Suppose I want to end up with \$5000 after 3 years and I can get 6% interest, compounded monthly. How much money would I have to put into the account to end up with \$5000?? (12)(3) .06 5000 1 12 P = + ( 29 36 5000 1 .005 P = + ( 29 36 5000 1.005 P = ( 29 36 5000 1.005 P = 4178.22 P = So yes we can find P. But suppose I only have \$4000 to put in savings. That will not grow to \$5000 in 3 years at 6%. So I need to find an account giving better than a 6% rate. What rate do I need?? (12)(3) 5000 4000 1 12 r = + can we find r?? 36 5000 1 4000 12 r = + 36 1.25 1 12 r = + 36 1.25 1 12 r = + 36 1.25 1 12 r - = 36 12( 1.25 1) r - = .0746 r = So I need about 7.5% interest rate.

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Now suppose I want to put \$4000 in an account and have it grow to \$5000 and the rate is 6% compounded monthly. Question is how long will it take to grow to \$5000. What is the value of “t”?? (12) .06 5000 4000 1 12 t = + Can we find “t”?? ( 29 12 5000 4000 1.005 t = ( 29 12 5000 1.005 4000 t = ( 29 12 1.25 1.005 t = Now, what can we do to get t by itself??? Similar to asking, in 17 3 x = what can we do to get x by itself ??? We have nothing at our command to solve this equation!! We can’t add something to both sides, subtract something on both sides, multiply, divide, take a root, raise to a power - - - nothing helps. We need help!! And help is
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Mth141s53 - Sec 5.3 Logarithms and Their Properties First...

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