Question

Solve each equation. $$ \log _{3} 243=2 x+1 $$

Short answer

x = 2

Step-by-step solution

Rewrite the Logarithmic Equation in Exponential Form

To solve for x, first rewrite the logarithmic equation \(\log _{3} 243=2 x+1\) in its exponential form. This means converting the logarithmic statement into \[3^{(2x+1)} = 243\].

Express Number in Terms of the Logarithm Base

Next, express 243 as a power of 3. Since \[243 = 3^5\], rewrite the equation as \[3^{(2x+1)} = 3^5\].

Equate the Exponents

Since the bases are the same, set the exponents equal to each other: \[2x + 1 = 5\].

Solve for x

Now, solve for x by isolating it. Subtract 1 from both sides: \[2x = 4\], then divide by 2: \[x = 2\].

Key concepts

exponential form

When solving logarithmic equations, it's often helpful to first convert the logarithmic equation to its exponential form. This means understanding the relationship between logarithms and exponents. For example, the logarithmic equation \(\text{log}_{3} 243 = 2x + 1\) can be rewritten in exponential form as \[3^{(2x+1)} = 243.\] By converting to this form, we can more easily manipulate and solve the equation. Remember, the general rule for converting a logarithm \[\log_b(a) = c\] to exponential form is \[b^c = a.\] This technique is useful because it translates the equation into a format that may be more straightforward to solve.

logarithms

Logarithms are the inverse operations of exponents. They help us solve for exponents in equations where the variable is in the exponent. For example, \(\text{log}_{3} 243 = 2x + 1\) is asking the question: 'To what power must 3 be raised to equal 243?' Knowing that \(\text{log}\_b (a) = c\) translates to \[b^c = a,\] we can convert this to the exponential form \[3^{(2x+1)} = 243.\] Essentially, logarithms simplify the process of finding the unknown exponent in such equations. This step is a crucial tool for solving equations where the variable appears in the exponent.

solving equations

Once we have our logarithmic equation in exponential form, solving it becomes a matter of algebra. For \(\text{log}_{3} 243 = 2x + 1\), converting to exponential form gives us \[3^{(2x+1)} = 243.\] We then express 243 as a power of the same base, which is 3. Noting that \[243 = 3^5,\] we set up the equation \[3^{(2x+1)} = 3^5.\] From here, because the bases are identical, we can equate the exponents directly: \[2x + 1 = 5.\] This eliminates the exponent and turns the problem into a simple linear equation. Subtracting 1 from both sides, we get \[2x = 4.\] Finally, dividing by 2, we solve for x: \[x = 2.\]

exponents

Exponents are a way of expressing repeated multiplication. For example, \[3^5 \] means multiplying 3 together five times: \[3 \times 3 \times 3 \times 3 \times 3 = 243.\] Understanding exponents is key to solving exponential and logarithmic equations. By knowing that \[3^{(2x+1)} = 243\] can be written as \[3^{(2x+1)} = 3^5,\] we make the equation simpler to handle. This equivalence allows us to equate the exponents directly since the bases are the same. Thus, understanding and manipulating exponents forms the foundation for solving these types of equations. By practicing exponent rules and operations, you can enhance your problem-solving skills for such logarithmic and exponential equations.