Recent questions tagged exponents

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Compute, if possible, the limits: \( \lim _{x \rightarrow 0} \sin (x) \) and \( \lim _{x \rightarrow 0} \sin (1 / x) \)
80
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Evaluate the following limit.\[\lim _{x \rightarrow \infty} x^{\frac{1}{x}}\]
84
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Evaluate $$ \lim_{x \to 0} \left(\frac{\sin(x)}{x}\right)$$
108
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Evaluate $\lim_{x \to \infty} \left(\frac{2x^2 - 3x + 1}{5x^2 + 2x - 1}\right) $
97
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Evaluate \( \lim_{x \to 2} \left(\frac{x^2 - 4}{x - 2}\right)\)
66
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Evaluate \(\lim_{x \to 2} (3x + 4) \)
96
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Find \( \lim _{x \rightarrow \pm \infty} \frac{7 x^{3}+2 x^{2}}{4 x^{3}-x} \).
128
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Evaluate \( \lim _{x \rightarrow 4^{+}} \frac{3}{x-4} \) and \( \lim _{x \rightarrow 4^{-}} \frac{3}{x-4} \)
89
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Find \( \lim _{x \rightarrow 0^{+}} \frac{|x|}{x} \) and \( \lim _{x \rightarrow 0^{-}} \frac{|x|}{x} \).
84
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Is \(\log_{10}2\) rational?
64
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Solve: \( \sqrt{\ln x}=\ln (\sqrt{x}) \).
57
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Find the derivative of the given function:\[y=\ln (\ln x)\]
66
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Solve \( \ln _{2}(x+1)-\ln _{2}(x-1)=\ln _{2} 8 \)
66
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Solve \( \ln _{3} x=5 \)
164
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Is the number \(-\frac{10}{3}\) a rational number?
127
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Evaluate the limit\[\lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5}\]
552
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What is the power rule for differentiation?
208
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Simplify \(\frac{3^{2 x} \cdot 27^{x-1}}{9^{2 x+2}}\)
274
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Write down the six laws for exponents. Use latex
367
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Convert \(\sqrt{49}=7\) into a logarithmic equation.
226
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Convert \(10^{-3}=\frac{1}{1,000}\) into a logarithmic equation.
213
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Rewrite \(5^3=125\) as a logarithmic equation.
383
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Convert \(\log _3\left(\frac{1}{81}\right)=-4\) into an exponential equation.
231
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Convert \(\log _4\left(\frac{1}{16}\right)=-2\) into an exponential equation.
550
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Rewrite \(\log _3 9=2\) as an exponential equation.
213
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Evaluate. \(x^3\) if \(x=-4\)
267
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Evaluate. \((5 x)^3\) if \(x=4\)
568
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What is a rational number?
258
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Solve for \(x\) \[ 4 x^2-4 x-1=0 \] (leave your answers in simplest surd form)
336
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How do you simplify an expression with exponents?
255
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Simplify the following:(a) \(\dfrac{\cos \left(180^{\circ}-\theta\right) \cdot \cos \left(90^{\circ}-\theta\right)}{\sin \left(90^{\circ}+\theta\right) \cdot \sin \left(-...
318
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Simplify the following using the laws of logarithms:(a) \(\quad \log _9 27\)(b) \(\quad \log _3 2 \cdot \log _2 5 \cdot \log _5 9\)
264
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Prove that \(\log _a x=\frac{\log _b x}{\log _b a}\)
273
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simplify \(\log _3 \sqrt{27}\)
308
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Simplify \(\log _3 \frac{1}{27}\)
283
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simplify \(-\log 4-\log 25\)
578
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simplify \(\log 20+\log 30-\log 6+\log _4 1\)
337
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Simplify using laws of logs \(\log _3 27-\log _3 3\)
257
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Simplify \(\log _9 27+\log _9 3\)
486
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Prove that \(\log _a a=1\)
288
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Using the laws of logarithms, expand \(\log _4\left(\frac{5}{x}\right)\)
198
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Expand \(\log_{3}5x\)
282
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Prove that \(\log _a x^m=m \log _a x \quad(x>0, y>0,0<a<1\) or \(a>1)\)
398
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Prove that \(\log _a\left(\frac{x}{y}\right)=\log _a x-\log _a y \quad(x>0, y>0,0<a<1\) or \(a>1)\)
315
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Prove that \(\log _a(x \times y)=\log _a x+\log _a y \quad(x>0, y>0,0<a<1\) or \(a>1)\)
424
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Find the derivative of the function $f(x) = x^3 + 3x^2 - 7x + 2$.
398
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Find the derivative of the function $f(x) = \ln x$
208
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Find the derivative of the function $f(x) = x^3 + 2x^2 + 3x + 4$
317
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Use the properties of logarithms to simplify the expression log base 5 of 25 + log base 5 of 125.
243
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Simplify the expression log base 10 of 100.
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