What is log_{5}(x)<= log_{x}(5) ?

The solution to log_{5}(x)<= log_{x}(5) is 0<x<= 1/5 \lor 1<x<= 5

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log_{5}(x)<= log_{x}(5)

lo g_{5} (x) \leq lo g_{x} (5)

0 < x \leq \frac{1}{5} or 1 < x \leq 5

Interval Notation

(0, \frac{1}{5}] \cup (1, 5]

Decimal Notation

0 < x \leq 0.2 or 1 < x \leq 5

Solution steps

lo g_{5} (x) \leq lo g_{x} (5)

Move

lo g_{x} (5)

to the left side

lo g_{5} (x) - lo g_{x} (5) \leq 0

Apply log rules

lo g_{5} (x) - lo g_{5} (5^{\frac{1}{l o g _{5} ( x )}}) \leq 0

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

lo g_{5} (5) \frac{1}{lo g _{5} ( x )}

Apply log rule:

lo g_{a} (a) = 1

\frac{1}{lo g _{5} ( x )}

Let

u = lo g_{5} (x)

u - \frac{1}{u} \leq 0

u - \frac{1}{u} \leq 0 : u \leq - 1 or 0 < u \leq 1

u \leq - 1 or 0 < u \leq 1

Substitute back

u = lo g_{5} (x)

lo g_{5} (x) \leq - 1 or 0 < lo g_{5} (x) \leq 1

lo g_{5} (x) \leq - 1 : 0 < x \leq \frac{1}{5}

0 < lo g_{5} (x) \leq 1 : 1 < x \leq 5

Combine the intervals

0 < x \leq \frac{1}{5} or 1 < x \leq 5

8 - 22

3 x^{2} - 6 x - 10 = 0

(10 - 5) (10 + 2)

4 x^{2} = 52

0.01 (\frac{55 0 ^{3}}{3} - \frac{50 0 ^{3}}{3})

What is log_{5}(x)<= log_{x}(5) ?
The solution to log_{5}(x)<= log_{x}(5) is 0<x<= 1/5 \lor 1<x<= 5