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Logarithm

Posted June 16th, 2009 by alpha
in

Find the value of $ log_7 \sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7.......infinity}}}}} $
Please tell me how to simplify $ \sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7.......infinity}}}}} $.
Kindly show your each step.

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Comments

answer

On June 16th, 2009 Structure says:
$$\sqrt{7}=7^{\frac{1}{2}}$$
$$\sqrt{7\sqrt{7}}=7^{\frac{1}{2}}7^{\frac{1}{2^2}}$$
$$\sqrt{7\sqrt{7\sqrt{7}}}=7^{\frac{1}{2}}7^{\frac{1}{2^2}}7^{\frac{1}{2^3}}=<br /> 7^{\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}}$$
$$x+x^2+x^3+...x^n=x\frac{1-x^n}{1-x}$$

For n factors
you have

$$7^{\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....\frac{1}{2^n}}=7^{\frac{1}{2}\frac{1-\frac{1}{2^n}}{1-\frac{1}{2}}}$$

As

$$\lim_{n \to \infty}\frac{1}{2^n}=0$$

your expression has value 7 and

$$\log_{7}7=1$$

~~THANK YOU ~~

On June 16th, 2009 alpha says:

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