Question

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

Find the value of z​

Answer 1

Identities :-

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

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

⠀ ━━━━━━━━━━━━━━━━━━━━━⠀⠀

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

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

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

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

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

$:\implies\sf{\log (abc)=\log z}$

$\sf{:\implies\boxed{\red{z=abc}}}$