Thread: How do you solve the following equation without using logarithms

1. How do you solve the following equation without using logarithms

27^4x = 81 x (1/9)^7x

2. Re: How do you solve the following equation without using logarithms

Write for example:
27^(4x) as 3^(12x)

Do this also for 81 and (1/9)^(7x)
So wright all the factors with the same base: 3

3. Re: How do you solve the following equation without using logarithms

Originally Posted by HelenC
27^4x = 81 x (1/9)^7x
You should use the fact that 9, 27 and 81 are all "powers of 3".

4. Re: How do you solve the following equation without using logarithms

Is this right so far?
27^4x = 81 x (1/9)^7x

27 = 3^3 so 27^(4x) = 3^(3*4x) = 3^(12x)
81 = 9^2
1/9 = 1/3^2

so 3^(12x) = (9^2) x (1/3^2)^7x

5. Re: How do you solve the following equation without using logarithms

putting everything in power of 3 - then
3^(12x) = 3^4 x (1/3^2)^(7x)

can you help me with what happens now?

6. Re: How do you solve the following equation without using logarithms

1) $\displaystyle 1/a^n= a^{-n}$

2) $\displaystyle (a^n)( a^m)= a^{m+n}$

3) $\displaystyle \left(a^m)^n= a^{mn}$

4) If $\displaystyle a^n= a^m$ then n= m.

Corrected thanks to ModusPonens

7. Re: How do you solve the following equation without using logarithms

Originally Posted by HelenC
putting everything in power of 3 - then
3^(12x) = 3^4 x (1/3^2)^(7x)

can you help me with what happens now?
$\displaystyle 3^{12x}=3^4\left[\frac{1}{\left(3^2\right)^{7x}}\right]$

Next obtain a single "power of 3" on the right, so that this power must equal 12x.

Alternatively, multiply both sides by $\displaystyle \left(3^2\right)^{7x}$

8. Re: How do you solve the following equation without using logarithms

Just a small correction: point 2) should be a product and not a sum.

9. Re: How do you solve the following equation without using logarithms

Nowadays, Logarithms ARE Exponents. Let's see you solve it without exponents. :-)

10. Re: How do you solve the following equation without using logarithms

To do it without log, make the base same if the variable is in the exponent. Make the exponents same, if the variable is in the base. Then equate the variable parts of both sides.
Otherwise log is the only way.

11. Re: How do you solve the following equation without using logarithms

Originally Posted by HelenC
27^4x = 81 x (1/9)^7x
Okay, no logs.

I astutely observe that 27, 81, and 1/9 are all integer powers of 3. I can transforme the equation:

$\displaystyle (3^{3})^{4x} = 3^{4}\cdot \left[(3^{-2})\right]^{7x}$

Using properties of exponents, I can transform the equatino further.

$\displaystyle 3^{12x} = 3^{4-14x}$

Observing that if the bases are equal, and the "=" is telling the truth, the exponents must be equal.

12x = 4-14x

26x = 4

Division

x = 4/26 = 2/13

Okay, since it was previously established that logarithms are the "only way" after that, let's try something else...

I type the whole thing into MathCad. Select and 'x' and pick "Solve". Bam!! x = 2/13. It may have used logarithms, I suppose.

Maybe a nice chart.

Yes, that looks like right around x = 2/13.

Some questions just challenge the purpose of learning or teaching? We HAVE logarithms! Why avoid them? There are MANY ways to proceed. Why state that there are only two ways?

Let's keep our minds open a bit, shall we?