Finite Math Examples

$5^{6 a - 3} = 2^{4 a + 1}$

Step 1

Take the log of both sides of the equation.

$\ln (5^{6 a - 3}) = \ln (2^{4 a + 1})$

Step 2

Expand $\ln (5^{6 a - 3})$ by moving $6 a - 3$ outside the logarithm.

$(6 a - 3) \ln (5) = \ln (2^{4 a + 1})$

Step 3

Expand $\ln (2^{4 a + 1})$ by moving $4 a + 1$ outside the logarithm.

$(6 a - 3) \ln (5) = (4 a + 1) \ln (2)$

Step 4

Solve the equation for $a$ .

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Step 4.1

Simplify $(6 a - 3) \ln (5)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (6 a - 3) \ln (5) = (4 a + 1) \ln (2)$

Step 4.1.2

Simplify by adding zeros.

$(6 a - 3) \ln (5) = (4 a + 1) \ln (2)$

Step 4.1.3

Apply the distributive property.

$6 a \ln (5) - 3 \ln (5) = (4 a + 1) \ln (2)$

Step 4.2

Simplify $(4 a + 1) \ln (2)$ .

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Step 4.2.1

Apply the distributive property.

$6 a \ln (5) - 3 \ln (5) = 4 a \ln (2) + 1 \ln (2)$

Step 4.2.2

Multiply $\ln (2)$ by $1$ .

$6 a \ln (5) - 3 \ln (5) = 4 a \ln (2) + \ln (2)$

Step 4.3

Subtract $4 a \ln (2)$ from both sides of the equation.

$6 a \ln (5) - 3 \ln (5) - 4 a \ln (2) = \ln (2)$

Step 4.4

Add $3 \ln (5)$ to both sides of the equation.

$6 a \ln (5) - 4 a \ln (2) = \ln (2) + 3 \ln (5)$

Step 4.5

Factor $2 a$ out of $6 a \ln (5) - 4 a \ln (2)$ .

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Step 4.5.1

Factor $2 a$ out of $6 a \ln (5)$ .

$2 a (3 \ln (5)) - 4 a \ln (2) = \ln (2) + 3 \ln (5)$

Step 4.5.2

Factor $2 a$ out of $- 4 a \ln (2)$ .

$2 a (3 \ln (5)) + 2 a (- 2 \ln (2)) = \ln (2) + 3 \ln (5)$

Step 4.5.3

Factor $2 a$ out of $2 a (3 \ln (5)) + 2 a (- 2 \ln (2))$ .

$2 a (3 \ln (5) - 2 \ln (2)) = \ln (2) + 3 \ln (5)$

Step 4.6

Divide each term in $2 a (3 \ln (5) - 2 \ln (2)) = \ln (2) + 3 \ln (5)$ by $2 (3 \ln (5) - 2 \ln (2))$ and simplify.

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Step 4.6.1

Divide each term in $2 a (3 \ln (5) - 2 \ln (2)) = \ln (2) + 3 \ln (5)$ by $2 (3 \ln (5) - 2 \ln (2))$ .

$\frac{2 a (3 \ln (5) - 2 \ln (2))}{2 (3 \ln (5) - 2 \ln (2))} = \frac{\ln (2)}{2 (3 \ln (5) - 2 \ln (2))} + \frac{3 \ln (5)}{2 (3 \ln (5) - 2 \ln (2))}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $2$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{2 a (3 \ln (5) - 2 \ln (2))}{2 (3 \ln (5) - 2 \ln (2))} = \frac{\ln (2)}{2 (3 \ln (5) - 2 \ln (2))} + \frac{3 \ln (5)}{2 (3 \ln (5) - 2 \ln (2))}$

Step 4.6.2.1.2

Rewrite the expression.

$\frac{a (3 \ln (5) - 2 \ln (2))}{3 \ln (5) - 2 \ln (2)} = \frac{\ln (2)}{2 (3 \ln (5) - 2 \ln (2))} + \frac{3 \ln (5)}{2 (3 \ln (5) - 2 \ln (2))}$

Step 4.6.2.2

Cancel the common factor of $3 \ln (5) - 2 \ln (2)$ .

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Step 4.6.2.2.1

Cancel the common factor.

$\frac{a (3 \ln (5) - 2 \ln (2))}{3 \ln (5) - 2 \ln (2)} = \frac{\ln (2)}{2 (3 \ln (5) - 2 \ln (2))} + \frac{3 \ln (5)}{2 (3 \ln (5) - 2 \ln (2))}$

Step 4.6.2.2.2

Divide $a$ by $1$ .

$a = \frac{\ln (2)}{2 (3 \ln (5) - 2 \ln (2))} + \frac{3 \ln (5)}{2 (3 \ln (5) - 2 \ln (2))}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{\ln (2)}{2 (3 \ln (5) - 2 \ln (2))} + \frac{3 \ln (5)}{2 (3 \ln (5) - 2 \ln (2))}$

Decimal Form:

$a = 0.80206708 \dots$