Question

If x=loga base 2a,y=log2a base 3a,z=log 3a base 4a.Show that xyz+1= 2yz

Answer 1

Answer:

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Answer 2

$\underline\mathfrak{Given:-}$

• x=loga base 2a,y=log2a base 3a,z=log 3a base 4a.

$\underline\mathfrak{To \: \: Find:-}$

• Show that xyz+1= 2yz ......?

$\underline\mathfrak{Solutions:-}$

• $\: \: \: \: \: {x} \: \: = \: \: {log_{2a}} \: \: \: \: {a} \: \: \leadsto \: \: a \: \: = \: \: {(2a)}^{x}$

On taking log on both side, we have.

$\: \: \: \: \: \leadsto \: \: {log} \: {a} \: \: = \: \: {x} \: {log} \: {2a}$

$\: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: \frac{log \: {a}}{log \: {2a}}$

• $\: \: \: \: \: {x} \: \: = \: \: {log_{3a}} \: \: \: \: {2a} \: \: \leadsto \: \: {2a} \: \: = \: \: {(3a)}^{x}$

On taking log on both side, we have.

$\: \: \: \: \: \leadsto \: \: {log} \: {2a} \: \: = \: \: {x} \: {log} \: {3a}$

$\: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: \frac{log \: {2a}}{log \: {3a}}$

• $\: \: \: \: \: {x} \: \: = \: \: {log_{4a}} \: \: \: \: {3a} \: \: \leadsto \: \: {3a} \: \: = \: \: {(4a)}^{x}$

On taking log on both side, we have.

$\: \: \: \: \: \leadsto \: \: {log} \: {3a} \: \: = \: \: {x} \: {log} \: {4a}$

$\: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: \frac{log \: {3a}}{log \: {4a}}$

$\: \: \: \: \: \therefore \: \: xyz \: \: = \: \: {\frac{log \: {a}}{log \: {2a}}} \: \times \: {\frac{log \: {2a}}{log \: {3a}}} \: \times \: {\frac{log \: {3a}}{log \: {4a}}}$

$\: \: \: \: \: \leadsto \: \: xyz \: \: = \: \: {\frac{log \: {a}}{log \: {4a}}}$

Now,

$\: \: \: \: \: \leadsto \: \: xyz \: + \: {1}$

$\: \: \: \: \: \leadsto \: \: \frac{log \: {a}}{log \: {4a}} \: + \: {1}$

$\: \: \: \: \: \leadsto \: \: \frac{log \: {a} \: + \: {log \: {4a}}}{log \: {4a}}$

$\: \: \: \: \: \leadsto \: \: \frac{log \: {4a}^{2}}{log \: {4a}}$

$\: \: \: \: \: \leadsto \: \: \frac{log \: {(2a)}^{2}}{log \: {4a}}$

$\: \: \: \: \: \leadsto \: \: \frac{{2} \: log \: {2a}^{2}}{log \: {4a}}$

$\: \: \: \: \: \leadsto \: \: \frac{{2} \: log \: {2a}^{2}}{log \: {3a}} \: \times \: \frac{log \: {3a}}{log \: {4a}}$

$\: \: \: \: \: \leadsto \: \: {2yz}$

So, the value of xyz + 1 is 2yz.