I am facing some trouble in solving this equation:
$$ 3\cdot x^{\log_5 2} + 2^{\log_5 x} = 64 $$
Give me some hints to proceed on this.
I am facing some trouble in solving this equation:
$$ 3\cdot x^{\log_5 2} + 2^{\log_5 x} = 64 $$
Give me some hints to proceed on this.
HINT $\rm\ \ x^{\: log_5 2}\ =\ 5^{\: log_5 x\ log_5 2}\ =\ 2^{\: log_5 x}\ \ $ (or take $\rm\:log_5\:$ of both sides if that's clearer to you)
Hence the equation reduces to $\rm\ 2^{\: \log_5 x}\ =\ 2^4\ \Rightarrow\ log_5\: x\ =\ 4\ \Rightarrow\ x\ =\ \ldots$
TIP $\ $ Typically such exponential equations will be solvable in closed form only if the exponentials are all linearly dependent, so you should always check for that first.
Hint: 3y + y = 64
Further (basic) hint: $a^{\log _b c} = (e^{\log a} )^{\log c/\log b} $