# Thread: Simple question; solving a logarithmic equation without calc.

1. ## Simple question; solving a logarithmic equation without calc.

Hi! I've not done any mathematics in over a year and am soon to be studying again. I'm slightly embarrassed to ask such a basic question.

5-2*14^3.2x=4

How do I solve this without the use of a calculator? If it can't be solved that way, what's the most simplified form of it?

2. ## Re: Simple question; solving a logarithmic equation without calc.

hi,
i also mentioned this que. many times but no body give right answers if you find the answer please tell me and i will also try to find our answer.
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vedicmaths

3. ## Re: Simple question; solving a logarithmic equation without calc.

Hi! I've not done any mathematics in over a year and am soon to be studying again. I'm slightly embarrassed to ask such a basic question.

5-2*14^3.2x=4

How do I solve this without the use of a calculator? If it can't be solved that way, what's the most simplified form of it?
Hej,

I'm guessing now: You are asked to solve

$\displaystyle 5-2 \cdot 14^{3.2 \cdot x} = 4$

If and only if this is correct rewrite the equation to

$\displaystyle 14^{3.2 \cdot x} = \frac12$

Now use logarithms:

$\displaystyle x = \frac1{3.2} \cdot \log_{14}\left(\frac12 \right)$

Use the base-change-formula:

$\displaystyle x = \frac{10}{32} \cdot \frac{\ln\left(\frac12 \right)}{\ln(14)}$

Simplify!

For confirmation only: You should come out with: $\displaystyle x = - \frac{5 \cdot \ln(2)}{16 \cdot \ln(14)} \approx -0.0821$

4. ## Re: Simple question; solving a logarithmic equation without calc.

The basic strategy for solving exponent problems: use ordinary algebra to get the exponent on one side and everything else on the other. Then use logs to get the x (or in this case the 3.2x) out of the exponent. Finally, do more algebra to finish solving for x.

In this case, the equation is 5-2*(the exponent part)=4. So subtract five and divide by -2 to get "the exponent part" on the left. You then have 14^(3.2x) on the left and everything else on the right. Rewrite as a log: the base of the exponent is the base of the log, the old answer becomes the target of the log and the old exponent becomes the new answer. Now you have the log to base 14 of one half equaling 3.2x. Divide by 3.2 to isolate x.

You now have what we call the "exact answer," ie, an answer with a log still in it. If you need a decimal number, you would have to plug your expression into a calculator at this point. This is the "approximate answer," because the decimal goes on forever and we have to round it off. (In order to plug your answer into some calculators, you might need to use the Change of Base formula to rewrite log to base 14 of 1/2 as log of 1/2 divided by log of 14.)

5. ## Re: Simple question; solving a logarithmic equation without calc.

5 - 2*14^3.2x = 4
subtract 5 from both sides & change signs
2*14^3.2x = 1
divide by 2
14^3.2x = 1/2
Take logs of both sides
log(14^3.2x) = log (1/2)
using power log formula (log a^b = b log a)
3.2x * log 14 = log 0.5
replacing log (0.5) with - log 2 (as log (b/a) = - log(a/b))
3.2x * log 14 = - log 2
dividing by 3.2 log 14
x = - log 2/(3.2 log 14)
= -5 log 2 / 16 log 14
= - 0.082 (calculator needed here at this last step)

6. ## Re: Simple question; solving a logarithmic equation without calc.

If you don't have a calculator use a table of logarithms!

7. ## Re: Simple question; solving a logarithmic equation without calc.

Gosh! Thank you all so much for the quick response, Thank you for solving that pesky problem.