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Логаритамске једначине 5

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Решити логаритамску једначину.

пр.11)   ${2^{2\lg 4x - 1}} - {7^{\lg 4x}} = {7^{\lg 4x - 1}} - 3 \cdot {4^{\lg 4x}}$

пр.12)   ${x^{\lg x}} = 1000{x^2}$

Пр.11

\[\begin{gathered}
{2^{2\lg 4x - 1}} - {7^{\lg 4x}} = {7^{\lg 4x - 1}} - 3 \cdot {4^{\lg 4x}} \hfill \\
4x > 0 \Rightarrow x > 0 \hfill \\
{2^{2\lg 4x - 1}} + 3 \cdot {4^{\lg 4x}} = {7^{\lg 4x - 1}} + {7^{\lg 4x}} \hfill \\
{2^{2\lg 4x}} \cdot {2^{ - 1}} + 3 \cdot {2^{2\lg 4x}} = {7^{\lg 4x}} \cdot {7^{ - 1}} + {7^{\lg 4x}} \hfill \\
{2^{2\lg 4x}} \cdot \left( {{2^{ - 1}} + 3} \right) = {7^{\lg 4x}} \cdot \left( {{7^{ - 1}} + 1} \right) \hfill \\
{2^{2\lg 4x}} \cdot \left( {\frac{7}{2}} \right) = {7^{\lg 4x}} \cdot \left( {\frac{8}{7}} \right)\left| { \div {7^{\lg 4x}}} \right. \hfill \\
\frac{{{2^{2\lg 4x}}}}{{{7^{\lg 4x}}}} \cdot \left( {\frac{7}{2}} \right) = \left( {\frac{8}{7}} \right)\left| { \cdot \frac{2}{7}} \right. \hfill \\
\frac{{{2^{2\lg 4x}}}}{{{7^{\lg 4x}}}} = \frac{8}{7} \cdot \frac{2}{7} \hfill \\
\frac{{{4^{\lg 4x}}}}{{{7^{\lg 4x}}}} = \frac{{16}}{{49}} \hfill \\
{\left( {\frac{4}{7}} \right)^{\lg 4x}} = {\left( {\frac{4}{7}} \right)^2} \hfill \\
\lg 4x = 2 \hfill \\
4x = {10^2} \hfill \\
4x = 100 \hfill \\
x = 25 \hfill \\
\end{gathered} \]

 

Пр.12

\[\begin{gathered}
{x^{\lg x}} = 1000{x^2} \hfill \\
x > 0 \hfill \\
\lg {x^{\lg x}} = \lg 1000{x^2} \hfill \\
\lg x \cdot \lg x = \lg 1000 + \lg {x^2} \hfill \\
{\left( {\lg x} \right)^2} = \lg {10^3} + \lg {x^2} \hfill \\
{\left( {\lg x} \right)^2} = 3 + 2\lg x \hfill \\
смена:\lg x = t \hfill \\
{t^2} - 2t - 3 = 0 \hfill \\
{t_{1,2}} = \frac{{2 \pm \sqrt {4 + 12} }}{2} \hfill \\
{t_{1,2}} = \frac{{2 \pm 4}}{2} \hfill \\
{t_1} = - 1,{t_2} = 3 \hfill \\
\lg x = - 1;\lg x = 3 \hfill \\
\hfill \\
\lg x = - 1 \hfill \\
x = {10^{ - 1}} = \frac{1}{{10}} \hfill \\
\hfill \\
\lg x = 3 \hfill \\
x = {10^3} = 1000 \hfill \\
x = \left\{ {\frac{1}{{10}};1000} \right\} \hfill \\
\end{gathered} \]

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