Solve 2log3x-4logx27<=5 - ScanMath

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Question

Solve the inequality

x \geq \frac{10}{243000}

Alternative Form

x \in [\frac{10}{243000}, + \infty)

Evaluate

2 lo g_{10} (3 x) - 4 lo g_{10} (x \times 27) \leq 5

Find the domain

More Steps

2 lo g_{10} (3 x) - 4 lo g_{10} (x \times 27) \leq 5, x > 0

Simplify

More Steps

2 lo g_{10} (\frac{1}{243 x}) \leq 5

Divide both sides

\frac{2 lo g _{10} ( \frac{1}{243 x} )}{2} \leq \frac{5}{2}

Divide the numbers

lo g_{10} (\frac{1}{243 x}) \leq \frac{5}{2}

For 10 >1 the expression lo g_{10} (\frac{1}{243 x}) \leq \frac{5}{2} is equivalent to \frac{1}{243 x} \leq 1 0^{\frac{5}{2}}

\frac{1}{243 x} \leq 1 0^{\frac{5}{2}}

Evaluate the power

\frac{1}{243 x} \leq 10010

Calculate

\frac{1}{243 x} - 10010 \leq 0

Calculate

More Steps

\frac{1 - 24300 10 \times x}{243 x} \leq 0

Separate the inequality into 2 possible cases

{1 - 2430010 \times x \geq 0 243 x < 0 {1 - 2430010 \times x \leq 0 243 x > 0

Solve the inequality

More Steps

{x \leq \frac{10}{243000} 243 x < 0 {1 - 2430010 \times x \leq 0 243 x > 0

Solve the inequality

{x \leq \frac{10}{243000} x < 0 {1 - 2430010 \times x \leq 0 243 x > 0

Solve the inequality

More Steps

{x \leq \frac{10}{243000} x < 0 {x \geq \frac{10}{243000} 243 x > 0

Solve the inequality

{x \leq \frac{10}{243000} x < 0 {x \geq \frac{10}{243000} x > 0

Find the intersection

x < 0 {x \geq \frac{10}{243000} x > 0

Find the intersection

x < 0 x \geq \frac{10}{243000}

Find the union

x \in (- \infty, 0) \cup [\frac{10}{243000}, + \infty)

Check if the solution is in the defined range

x \in (- \infty, 0) \cup [\frac{10}{243000}, + \infty), x > 0

Solution

x \geq \frac{10}{243000}

Alternative Form

x \in [\frac{10}{243000}, + \infty)

Show Solution