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quarto/precalc/exp_log_functions.qmd

@@ -27,7 +27,7 @@ The family of exponential functions is defined by $f(x) = a^x, -\infty< x < \inf
 For a given $a$, defining $a^n$ for positive integers is straightforward, as it means multiplying $n$ copies of $a.$ From this, for *integer powers*, the key properties of exponents: $a^x \cdot a^y = a^{x+y}$, and $(a^x)^y = a^{x \cdot y}$ are immediate consequences. For example with  $x=3$ and $y=2$:
 
 
-$$
+
 \begin{align*}
 a^3 \cdot a^2 &= (a\cdot a \cdot a) \cdot (a \cdot a) \\
               &= (a \cdot a \cdot a \cdot a \cdot a) \\
@@ -36,7 +36,7 @@ a^3 \cdot a^2 &= (a\cdot a \cdot a) \cdot (a \cdot a) \\
         &= (a\cdot a \cdot a \cdot a\cdot a \cdot a) \\
 		&= a^6 = a^{3\cdot 2}.
 \end{align*}
-$$
+
 
 For $a \neq 0$, $a^0$ is defined to be $1$.
 
@@ -388,13 +388,13 @@ In short, we have these three properties of logarithmic functions:
 If $a, b$ are positive bases; $u,v$ are positive numbers; and $x$ is any real number then:
 
 
-$$
+
 \begin{align*}
 \log_a(uv) &= \log_a(u) + \log_a(v),    \\
 \log_a(u^x) &= x \log_a(u), \text{ and} \\
 \log_a(u) &= \log_b(u)/\log_b(a).
 \end{align*}
-$$
+
 
 ##### Example