How to solve this equation?

Question

(2x+5)^(x+1)=729

x+1=ln(729)/ln(2x+5)

What do I do from here? All I know is that the answer is x=2. Please show all your work. Thanks.

Answer 1

substitute:

(2x+5)^(x+1)=9^3=729 x=2

x+1=6.59/ln(2x+5)

6.59=(x+1)ln(2x+5)

=3*ln9

=6.59

Answer 2

Use a change of base formula for the natural logs:

log_aN= lnN/lna so ln729/ln(2x+5) = log_2x+5729 which = x+1

Now put both sides of that equation as exponents of 2x+5 to get 729 = (2x+5)^x+1 which is your first equation. They're both equivalent. Neither one adds any new information. You have really one equation, one unknown, not two independent equations.

x=2 is clearly a solution, as you can see by plugging 2 in place of x into that equation 729=(2x+5)^x+1 as it reduces to 729 =(9)³ =27³ = 729

Any number greater than 2 will make the right side greater than 729. Try plugging in a few numbers, such as 3 or 4. It gets exponentially larger.

Any number less than 2 will make the right side less than 729.

Try plugging in a few simple integers, like 1, 0 or -1. It just gets smaller.

x=2 is the only solution.

That is often a good way to start, by plugging in some simple integers to narrow down the range of the solution. You try 0, 1 and 2 and you stumble upon the only answer.