Exponential Variable and Logarithms

In summary, I was trying to find the value of a variable which happens to be an exponent, but I'm stuck. I believe I need to use logarithms to get to my answer, but I've reviewed logarithm rules and I'm stuck. I've divided 230k by 1500, but am stuck at that point. Now I'm at 153.33, but I think I need to log both sides and I'm stuck. Any assistance would be greatly appreciated.
  • #1
jsully
7
0

Homework Statement



I'm trying to find the value of a variable which happens to be an exponent.

Homework Equations



[itex]230,000=1500\frac{(1+.00077)^n-1}{.00077}[/itex]

I believe I need to use logarithms to get to my answer, but I've reviewed logarithm rules and I'm stuck.

The Attempt at a Solution



I've divided 230k by 1500, but am stuck at that point.

I'm now at

[itex]153.33=\frac{(1+.00077)^n-1}{.00077}[/itex]

...and stuck :( I think I need to log both sides, something like [itex]log 153.33=(n)log \frac{(1+.00077)-1}{.00077}[/itex]
I've tried simplifying the right side, but I end up at 1 which means the right side ends up being zero, which doesn't make any sense.

Any assistance would be greatly appreciated.
 
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  • #2
Do you know how to solve ##x^n=y## for n? Can you rewrite the equation in that form?
 
  • #3
jsully said:

Homework Statement



I'm trying to find the value of a variable which happens to be an exponent.

Homework Equations



[itex]230,000=1500\frac{(1+.00077)^n-1}{.00077}[/itex]

I believe I need to use logarithms to get to my answer, but I've reviewed logarithm rules and I'm stuck.

The Attempt at a Solution



I've divided 230k by 1500, but am stuck at that point.

I'm now at

[itex]153.33=\frac{(1+.00077)^n-1}{.00077}[/itex]

...and stuck :( I think I need to log both sides, something like [itex]log 153.33=(n)log \frac{(1+.00077)-1}{.00077}[/itex]
No... you can't do that! Before taking the logarithm of both sides, I would isolate the (1+.00077)^n portion. Can you do that?
 
  • #4
Would it be [itex]n=\frac{log153.33}{log(1+.00077)-1}{.00077}[/itex]
 
Last edited:
  • #5
Yeah, nevermind that can't be right. Very frustrating..
 
  • #6
Fredrik said:
Do you know how to solve ##x^n=y## for n? Can you rewrite the equation in that form?

I mean, I know that if x^n=y then ln(y)/ln(x)=n. I can't figure out how to write using my values though.
 
  • #7
No. In addition to reviewing the rules about logarithms, you need to brush up on your algebra as well.

You had 153.33 = ((1+0.00077)^n - 1) / 0.00077

You can further simplify:

153.33(0.00077) = (1+0.00077)^n - 1
153.33(0.00077)+1 = (1+0.00077)^n

Now use logarithms:

log (153.33(0.00077) + 1) = n log (1+0.00077)

Therefore:

n = log(153.33(0.00077)+1) / log (1+0.00077)

or

n = 144.99
 
  • #8
You probably shouldn't give away the complete solution like that. A better hint would be to ask if jsully knows how to solve
$$a=b\frac{x-1}{c}$$ for x when a,b,c are non-zero real numbers.
 

FAQ: Exponential Variable and Logarithms

What is an exponential variable?

An exponential variable is a mathematical concept that involves a number (called the base) being multiplied by itself multiple times. The exponent, which is a small number written above and to the right of the base, determines the number of times the base is multiplied by itself. For example, 23 means 2 is multiplied by itself 3 times, resulting in 8.

What is a logarithm?

A logarithm is the inverse function of an exponential variable. It helps to determine the exponent in an exponential equation. For example, in the equation 2x = 16, the logarithm (base 2) of 16 is 4, so x = 4.

How are exponential variables and logarithms used in science?

Exponential variables and logarithms are used in various scientific fields, such as biology, chemistry, and physics. They are used to model and analyze data that follows exponential growth or decay patterns, such as population growth, radioactive decay, and chemical reactions.

What is the difference between natural logarithm and common logarithm?

The natural logarithm (ln) is the logarithm with base e, which is an irrational number approximately equal to 2.718. The common logarithm (log) is the logarithm with base 10. Both have different properties and are used in different contexts.

What are some real-life applications of exponential variables and logarithms?

Exponential variables and logarithms are used in various real-life applications, such as calculating interest rates in finance, predicting population growth in biology, measuring sound intensity in physics, and analyzing data in epidemiology. They are also used in technology, such as signal processing and computer algorithms.

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