$1500=P \times { (1 + 0.02) }^{ 24 }$, what is the value of $P$?

Question

Hey guys could you please tell me what is the faster why to solve this equation. It's a compound interest equation and I'm stuck at the ${ (1 + 0.02) }^{ 24 }$ I really don't know how to proceed in this part. I thought about log, but how can I apply logarithm without touching on the $1500$?

Any help is appreciated, thanks.

Answer 1

$1500=P \times { (1 + 0.02) }^{ 24 }$ if you don't want to touch $1500$ then just find out the value of ${ (1 + 0.02) }^{ 24 }$ .you can do it as in your selected answer and via log table like this: $$x={ (1 + 0.02) }^{ 24 }$$ take log on both side $$\log x=\log { (1 + 0.02) }^{ 24 }$$ $$\log x=24\log { (1.02) }$$ $$\log x=24\times 0.0086$$ $$\log x=0.2064$$ $$x=Antilog(0.2064)$$ antilog is a inverse function of $\log$ there is also an antilog table to see value.To calculate via calculator or manually it is like this: $$x=10^{0.2064}$$ $$x=1.6084$$ Now you have value of $(1.02)^{24}\;$which is $1.6084$ just put this value in equation : $$P=\dfrac{1500}{1.6084}\implies P=932.58$$

Answer 2

You have $P=\frac {1500}{1.02^{24}}$. To get $1.02^{24}$, you can use log tables to get $1.02^{24}=\exp(24 \ln 1.02)$. You can use repeated squaring to get$1.02^2, 1.04^4, \dots 1.02^{16}$ and finally use $1.02^{24}=1.02^{16}\cdot1.02^8$ which only takes five multiplies, or you can get a calculator that does powers. The final value is about $1.608$

Answer 3

To use a logarithm you would have to touch the 1500 [or 15,000, you reference both, we'll assume 1500]. It would look like

$$\log(1500)=24\times\log(P\times1.02)$$

Alternatively, if you wanted to approximate your solution you could use the binomial expansion with 0.02 as your small number and write something like

$$1500\approx P\times(1+24\times0.02)=P\times1.48$$

which is easily solved by hand. Otherwise using a calculator is your best option.