# Exponentiation Problem

$$60^{a} = 3$$

$$60^{b} = 5$$

$$12^{\frac{1 - a - b}{2(1 - b)}} = ?$$

I am just unable to solve it. So, any help would be highly appreciated.

Thanks.

Note by Nilabha Saha
2 years, 3 months ago

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$$a=\dfrac{\log3}{\log{60}}$$. $$b=\dfrac{\log5}{\log{60}}$$.

$$12^{\frac{1-(a+b)}{2(1-b)}}$$$$=12^{\frac{1-\log15/\log{60}}{2(1-\log5/{60})}}=$$$$12^{\frac{\log4/\log{60}}{2(\log{12}/\log{60})}}=$$$$12^{\log2/\log{12}}=12^{\log_{12}{2}}=2$$.

Note:

$$\log a+\log b=\log{ab}$$

$$\log a-\log b=\log{a/b}$$

$$\log a^2=2 \log a$$

$$a^{\log_a b}=b$$.

Here, we take $$1=\log{60}/\log{60}$$.

- 2 years, 3 months ago

Thank you very much.☺ Helped me understand a lot.😃😃😃

- 2 years, 3 months ago