Logarithmic and Exponential Equations

Solving Logarithmic Equations

To solve logarithmic equations we take advantage of the properties of logarithmic functions and the fact that

$y=log_{a}(x){\text{ is equivalent to }}x=a^{y}~a>0,~a\neq 1$

We also use the additional fact that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log_a(M) = log_a(N) } then M = N for M, a, N positive numbers and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \neq 1}

Example

Solve: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log_5(x+ 6) + log_5(x + 2) = 1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} log_5(x + 6) + log_5(x + 2) & = & 1\\ log_5( (x + 6)(x + 2)) & = & 1\\ (x + 6)(x + 2) & = & 5\\ x^2 + 8x + 12 & = & 5 \\ x^2 + 8x + 7 & = & 0\\ (x + 1)(x + 7) & = & 0 \end{array}}

Now we just need to make sure our answers make sense. When x = -7, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log_5(-1) + log_5(-5)} which cannot occur since the domain of the logarithm function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, \infty)}

Solving Exponential Equations

In a similar fashion to solving logarithmic equations, we can solve exponential equations by using their properties and the fact that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^u = a^v ~\text{ then } u = v~ a > 0, ~a \neq 1}

Example:

Solve: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8\cdot 3^x = 5}

We start by dividing both sides by 8 to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = \frac{5}{8}} . Taking the log base 3 of both sides we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log_3(3^x) = log_3(\frac{5}{8})} . Finally by our properties of logarithms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = log_3(\frac{5}{8})} .

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