Question

If 1/log_a t + 1/log_b t + 1/log_c t = 1/log_z t the value of z is ?

Please Give Full Solution for Q. 81

(a) abc is Correct Answer

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Answer 1

Let us recall the logarithmic identities,

$\longrightarrow\sf{\log_ba=\dfrac{\log a}{\log b}\quad\quad\dots(1)}$

$\longrightarrow\sf{\log a+\log b=\log(ab)\quad\quad\dots(2)}$

Given in the question,

$\longrightarrow\sf{\dfrac{1}{\log_at}+\dfrac{1}{\log_bt}+\dfrac{1}{\log_ct}=\dfrac{1}{\log_zt}}$

Using (1),

$\longrightarrow\sf{\dfrac{1}{\left(\dfrac{\log t}{\log a}\right)}+\dfrac{1}{\left(\dfrac{\log t}{\log b}\right)}+\dfrac{1}{\left(\dfrac{\log t}{\log c}\right)}=\dfrac{1}{\left(\dfrac{\log t}{\log z}\right)}}$

$\longrightarrow\sf{\dfrac{\log a}{\log t}+\dfrac{\log b}{\log t}+\dfrac{\log c}{\log t}=\dfrac{\log z}{\log t}}$

$\longrightarrow\sf{\dfrac{\log a+\log b+\log c}{\log t}=\dfrac{\log z}{\log t}}$

$\longrightarrow\sf{\log a+\log b+\log c=\log z}$

Using (2),

$\longrightarrow\sf{\log (abc)=\log z}$

Taking antilog,

$\sf{\longrightarrow\underline{\underline{z=abc}}}$

Hence (a) is the answer.