Solve for x: a) 3(2x + 1) = 3 b) 3(x + 5) − 3(x − 1) = 3 c) 33x = 1/729

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:

a) Solve for $x$ : $\log_3(2x + 1) = 3$

Convert the logarithmic equation to exponential form.
The definition of a logarithm states that if $\log_b y = x$ , then $b^x = y$ .
$3^3 = 2x + 1$
Calculate $3^3$ and solve for $x$ .
$27 = 2x + 1$ $27 - 1 = 2x$ $26 = 2x$ $x = \frac{26}{2}$ $x = 13$
Since $2(13) + 1 = 27 > 0$ , the solution is valid.
$\boxed{\mathbf{x = 13}}$
b) Solve for $x$ : $\log_3(x + 5) - \log_3(x - 1) = 3$
Use the logarithm property $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$ to combine the terms.
$\log_3\left(\frac{x + 5}{x - 1}\right) = 3$
Convert the logarithmic equation to exponential form.
$3^3 = \frac{x + 5}{x - 1}$ $27 = \frac{x + 5}{x - 1}$
Multiply both sides by $(x - 1)$ and solve for $x$ .
$27(x - 1) = x + 5$ $27x - 27 = x + 5$ $27x - x = 5 + 27$ $26x = 32$ $x = \frac{32}{26}$ $x = \frac{16}{13}$
Check for validity:
$x+5 = \frac{16}{13} + 5 = \frac{16+65}{13} = \frac{81}{13} > 0$
$x-1 = \frac{16}{13} - 1 = \frac{16-13}{13} = \frac{3}{13} > 0$
Both arguments are positive, so the solution is valid.
$\boxed{\mathbf{x = \frac{16}{13}}}$
c) Solve for $x$ : $3^{3x} = \frac{1}{729}$
Express 729 as a power of 3.
$3^1 = 3$
$3^2 = 9$
$3^3 = 27$
$3^4 = 81$
$3^5 = 243$
$3^6 = 729$
So, $729 = 3^6$ .
Rewrite the equation using the power of 3.
$3^{3x} = \frac{1}{3^6}$
Use the property $\frac{1}{a^n} = a^{-n}$ to rewrite the right side.
$3^{3x} = 3^{-6}$
Equate the exponents since the bases are the same.
$3x = -6$ $x = \frac{-6}{3}$ $x = -2$ $\boxed{\mathbf{x = -2}}$
:

i) Express the following expressions in their simplest form:

a) $\sqrt{12}$
Find the largest perfect square factor of 12.
The factors of 12 are 1, 2, 3, 4, 6, 12. The largest perfect square factor is 4.
$\sqrt{12} = \sqrt{4 \times 3}$
Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify.
$\sqrt{12} = \sqrt{4} \times \sqrt{3}$ $\sqrt{12} = 2\sqrt{3}$ $\boxed{\mathbf{2\sqrt{3}}}$
b) $\sqrt{50}$
Find the largest perfect square factor of 50.
The factors of 50 are 1, 2, 5, 10, 25, 50. The largest perfect square factor is 25.
$\sqrt{50} = \sqrt{25 \times 2}$
Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify.
$\sqrt{50} = \sqrt{25} \times \sqrt{2}$ $\sqrt{50} = 5\sqrt{2}$ $\boxed{\mathbf{5\sqrt{2}}}$
ii) Rationalize the following surds:

a) $\frac{2}{1+\sqrt{3}}$
Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of $1+\sqrt{3}$ is $1-\sqrt{3}$ .
$\frac{2}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}}$
Expand the numerator and denominator.
Use the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ for the denominator.
$\frac{2(1-\sqrt{3})}{1^2 - (\sqrt{3})^2}$ $\frac{2 - 2\sqrt{3}}{1 - 3}$ $\frac{2 - 2\sqrt{3}}{-2}$
Simplify the expression.
$\frac{2(1 - \sqrt{3})}{-2}$ $-(1 - \sqrt{3})$ $-1 + \sqrt{3}$ $\boxed{\mathbf{\sqrt{3} - 1}}$
b) $\frac{1}{\sqrt{5}-\sqrt{3}}$
Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of $\sqrt{5}-\sqrt{3}$ is $\sqrt{5}+\sqrt{3}$ .
$\frac{1}{\sqrt{5}-\sqrt{3}} \times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$
Expand the numerator and denominator.
Use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ for the denominator.
$\frac{1(\sqrt{5}+\sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2}$ $\frac{\sqrt{5}+\sqrt{3}}{5 - 3}$ $\frac{\sqrt{5}+\sqrt{3}}{2}$ $\boxed{\mathbf{\frac{\sqrt{5}+\sqrt{3}}{2}}}$

Solve for x: a) 3(2x + 1) = 3 b) 3(x + 5) − 3(x − 1) = 3 c) 33x = (1)/(729)

Step-by-step solution

Convert the logarithmic equation to exponential form.

Calculate $3^3$ and solve for $x$ .

Use the logarithm property $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$ to combine the terms.

Convert the logarithmic equation to exponential form.

Multiply both sides by $(x - 1)$ and solve for $x$ .

Express 729 as a power of 3.

Rewrite the equation using the power of 3.

Use the property $\frac{1}{a^n} = a^{-n}$ to rewrite the right side.

Equate the exponents since the bases are the same.

Find the largest perfect square factor of 12.

Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify.

Find the largest perfect square factor of 50.

Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify.

Multiply the numerator and denominator by the conjugate of the denominator.

Expand the numerator and denominator.

Simplify the expression.

Multiply the numerator and denominator by the conjugate of the denominator.

Expand the numerator and denominator.

Still stuck on this one?

More Mathematics Questions

Solve for x: a) 3(2x + 1) = 3 b) 3(x + 5) − 3(x − 1) = 3 c) 33x = (1)/(729)

Step-by-step solution

Convert the logarithmic equation to exponential form.

Calculate 333^333 and solve for xxx.

Use the logarithm property log⁡bM−log⁡bN=log⁡b(MN)\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)logb​M−logb​N=logb​(NM​) to combine the terms.

Convert the logarithmic equation to exponential form.

Multiply both sides by (x−1)(x - 1)(x−1) and solve for xxx.

Express 729 as a power of 3.

Rewrite the equation using the power of 3.

Use the property 1an=a−n\frac{1}{a^n} = a^{-n}an1​=a−n to rewrite the right side.

Equate the exponents since the bases are the same.

Find the largest perfect square factor of 12.

Use the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}ab​=a​×b​ to simplify.

Find the largest perfect square factor of 50.

Use the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}ab​=a​×b​ to simplify.

Multiply the numerator and denominator by the conjugate of the denominator.

Expand the numerator and denominator.

Simplify the expression.

Multiply the numerator and denominator by the conjugate of the denominator.

Expand the numerator and denominator.

Still stuck on this one?

More Mathematics Questions

Calculate $3^3$ and solve for $x$ .

Use the logarithm property $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$ to combine the terms.

Multiply both sides by $(x - 1)$ and solve for $x$ .

Use the property $\frac{1}{a^n} = a^{-n}$ to rewrite the right side.

Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify.

Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify.